Group Theory

Home , Group with operators, Hall subgroup, Holomorph (mathematics), Subgroup series

Hartmut Laue

Mathematisches Seminar der Universit¨at Kiel 2013

Preface

These lecture notes present the contents of my course on Group Theory within the masters programme in Mathematics at the University of Kiel. The aim is to introduce into concepts and techniques of modern group theory which are the prerequisites for tackling current research problems. In an area which has been studied with extreme intensity for many decades, the decision of what to include or not under the time limits of a summer semester was certainly not trivial, and apart from the aspect of importance also that of personal taste had to play a role. Experts will soon discover that among the results proved in this course there are certain theorems which frequently are viewed as too difficult to reach, like Tate’s (4.10) or Roquette’s (5.13). The proofs given here need only a few lines thanks to an approach which seems to have been underestimated although certain rudiments of it have made it into newer textbooks. Instead of making heavy use of cohomological or topological considerations or character theory, we introduce a completely elementary but rather general concept of normalized group action (1.5.4) which serves as a base for not only the above-mentioned highlights but also for other important theorems (3.6, 3.9 (Gaschütz), 3.13 (Schur-Zassenhaus)) and for the transfer. Thus we hope to escape the cartesian reservation towards authors in general1, although other parts of the theory clearly follow well-known patterns when a major modification would not result in a gain of clarity or applicability. Nevertheless, a closer look shows that details frequently differ from classical expositions. The reader is urged to consult these, to compare, to develop his own understanding and view. A major difference to traditional presentations will be observed with respect to the Krull-Schmidt theorem which is proved in a general version (2.5) containing as special cases the classical one (characterized by the hypothesis of chain conditions) and the form for modules due to Azumaya. Throughout these notes we avoid to suppose finiteness where not necessary. On the other hand, the spirit of the text is clearly determined by central developments in the theory of finite groups. The course does not start at the level of the definition of a group. Knowledge of certain basics, usually provided by a standard introductory course in Algebra, is assumed. But

1“[. . . ]Authors are ordinarily so disposed that whenever their heedless credulity has led them to a decision on some controverted opinion, they always try to bring us over to the same side, with the subtlest arguments; if on the other hand they have been fortunate enough to discover something certain and evident, they never set it forth without wrapping it up in all sorts of complications. I suppose they are afraid that a simple account may lessen the importance they gain by the discovery; or perhaps they begrudge us the plain truth.” Descartes, Reg. 3 (transl. E. Anscombe, P. T. Geach, Descartes Philosophical Writings, London 1954.)

1 apart from simplest standard topics, the course is self-contained. A detailed list of what will be assumed to be known may be found on p. 4. It is not necessary as a prerequisite to be informed about deeper properties of soluble groups and solubility criteria. These topics are essentially independent of and rather a supplement to these lecture notes, but of course highly recommended for a broader background. We think that a reader who feels attracted, hopefully even fascinated by groups will – and should – in any case study group rings, representations, characters, subjects which are not covered by the present text. The basics of these are treated, for example, in my course on solubility of equations and groups (currently running under the label “Algebra II” as part of the bachelor degree programme). Of course, there are more than enough specialized textbooks on these topics for readers who want to study them in greater detail.

Kiel, July 2013 Hartmut Laue

2 Contents

1 Permutations and group actions 5

2 Groups with operators 23

3 Complements 34

4 Transfer 48

5 Nilpotency 54

A Appendix and Outlook 68 Prerequisites

Readers should be familiar with the notions group, subgroup, normal subgroup, quotient (factor group), index, cyclic group, abelian group, soluble group, simple group, commutator, commutator subgroup of a group, symmetric group, sign homomorphism. It is assumed that the homomorphism theorem for groups is at the reader’s disposal, also the isomorphism theorems as consequences; in particular, it should be known how the subgroups of a factor group of G are obtained from subgroups of G. In general, the product set of two subgroups need not be a subgroup. But this is the case if one of the two subgroups is normal. The product of two normal subgroups is not only a subgroup but even normal. The order of a subgroup of a ﬁnite group G is a divisor of |G| (Lagrange’s theorem), and G is cyclic if and only if for each divisor d of |G| there exists exactly one subgroup of G of order d. A further most useful general proposition is the so-called extension principle a proof of which will not be given here: Let ϕ be an injective mapping of a set B into a set M. Then there exists a set Bˆ containing B and a bijection ϕˆ of Bˆ onto M such that ϕˆ|B = ϕ. If ◦ is an operation on B, an operation on M such that ϕ is a homomorphism with respect to these operations, then ◦ extends to an operation ◦ˆ of Bˆ such that ϕˆ is an isomorphism of (B,ˆ ◦ˆ) onto (M, ).

We write N for the set of all positive integers and set N0 := N ∪{0}. For all n ∈ N0 we put n := {k|k ∈ N, k ≤ n}. A dot above the symbol of a union ( ˙ ) is used if the union is disjoint. If G is a group, we write G′ for its commutator subgroup. Furthermore,  stands for the trivial subgroup {1G}.

4 1 Permutations and group actions

If X is a set and V is a set of operations on X, a bijection of X onto X which is a homomorphism with respect to each operation in V is called an automorphism of the structure (X, V). We write Aut (X, V) for the set of all automorphisms of (X, V). Recall that Aut (X, V) is a group with respect to the natural composition of mappings, called the automorphism group of (X, V). Usually it is clear from the context which operations on the set X are considered, i. e., there is no doubt about the set V: If we study groups, V will just contain a single element, the group operation; if we study ﬁelds, V will consist of the addition and the multiplication of the ﬁeld in question, etc. If there is no doubt about the meaning of V we speak simply of automorphisms of X and write Aut X for its automorphism group. Galois theory is a convincing and ground-laying example of the general idea to study structures by means of an analysis of their automorphism group. The trivial remark that the automorphisms of any structure always form a group explains an important universal aspect of the notion of group: Studying an arbitrary structure, describing the structural roles of its elements, leads and amounts to studying its automorphism group. This principle may even be applied to groups themselves as a special class of algebraic structures. Every element g of a group G induces a so-called inner automorphism g¯ of G by means of conjugation:

g¯ : G → G, x → xg := g−1xg, is an automorphism of G, and the mapping

κ : G → Aut G, g → g,¯ is a homomorphism whose kernel Z(G) is called the centre of G. We have the following 4-fold description of its elements:

g ∈ Z(G) ⇔ ∀x ∈ G xg = x ⇔ ∀x ∈ G xg = gx ⇔ ∀x ∈ G g = gx which allows the following three-fold interpretation of Z(G): 1.0.1. For every group G, Z(G) is the set of all elements of G – the conjugation by which induces the identity on G,

– which commute with every element of G,

– which are left ﬁxed under the conjugation by an arbitrary element of G.

5 In particular, Z(G) is an abelian normal subgroup of G. More generally, for every x ∈ G, the set Z(G) ∪{x} consists of mutually commuting elements, hence generates an abelian subgroup. If we assume that G/Z(G) is cyclic, we may choose x as an element of a generator of G/Z(G) and obtain: 1.0.2. If G/Z(G) is cyclic, G is abelian (i.e., G/Z(G)= ). For all g ∈ G, α ∈ Aut G we have2 gα = gα. Hence the image of G under κ is a normal subgroup of Aut G. It is denoted by In G and called the inner automorphism group of G. Thus, making use of the homomorphism theorem, we obtain

G/Z(G) ∼= In G Aut G for every group G.

A subset X of a group G is called normal if Xg = X for all g ∈ G.3 If X is a normal subset of G, then so is G X; in particular, G {1G} is normal. For every x ∈ G, the set xG := {xg|g ∈ G} is the smallest normal subset of G containing x and is called the conjugacy class of x in G. From the third description of Z(G) in 1.0.1 we obtain 1.0.3. For every element x of a group G, x ∈ Z(G) ⇔ xG = {x}⇔|xG| =1. Let ∼ be the relation on a group G defined by G x ∼ y ↔ ∃g ∈ G xg = y, for all x, y ∈ G. G 1.0.4. ∼ is an equivalence relation on G. The equivalence classes are exactly the con- G jugacy classes of G. In particular, the conjugacy classes of G form a partition of G. An important special case of an automorphism group Aut (X, V) arises if V = ∅ : We have Aut (X, ∅)= SX , the symmetric group on X, consisting of all bijections of X onto X. We shall now make several observations about our general notions in this special type of group. We confine ourselves to the case of a finite set X.

Let π ∈ SX . The relation ∼ on X deﬁned by π

m ∼ m′ ↔ ∃k ∈ Z mπk = m′, for all m, m′ ∈ X, π is an equivalence relation. In particular, its equivalence classes form a partition of X.

1.1 Deﬁnition. Let X be ﬁnite set and π ∈ SX . An equivalence class of ∼ is called a π π-orbit in X and nontrivial if it contains more than one element. If there is at most one nontrivial π-orbit, π is called a cycle.

2For the image of an element of a multiplicatively written group under a homomorphism, the expo- nential notation is convenient. This explains the meaning of gα while gα merely is a particular case of conjugation, within the group Aut G. 3equivalently, if Xg ⊆ X for all g ∈ G, i. e., if X is invariant under all conjugations.

6 Obviously, idX is the only permutation for which each orbit is trivial. Now let ζ ∈ SX 4 be a cycle = idX and m ∈ X such that mζ = m. If k is the order of ζ, we set

(∗) m1 := m, ∀j ∈ k {1} mj := mj−1ζ.

j−1 Then mj = mζ for all j ∈ k ; {m1,...,mk} is the nontrivial ζ-orbit and contains exactly k elements. By (∗), the k-tuple (m1,...,mk) determines uniquely the cycle ζ. It is customary to write the cycle ζ “abusing” the notation of an 1 × k matrix:

ζ = m1 ... mk

The choice of m = m1 was arbitrary within the nontrivial ζ-orbit so that any of the k elements mj could take its place. We have

m1 m2 ... mk = m2 ... mk m1 = = mk m1 ... mk−1 which, of course, makes sense only as series of equalities of cycles, not of 1 × k matrices. This notation is the reason for calling k the length of the cycle ζ. A cycle of length ′ k is called a k-cycle. Cycles ζ, ζ in SX are called disjoint if their nontrivial orbits (if existent) are disjoint. If this is the case, then each m ∈ X is ﬁxed by at least one of the two cycles. In particular, mζζ′ = mζ′ζ for all m ∈ X, i. e., ζ,ζ′ commute. We conclude: 1.1.1. Any composition of mutually disjoint cycles is independent of the choice of the order of the factors.

Thus, for any set C of mutually disjoint cycles = idX in SX , the product π = C is well-deﬁned. If m ∈ X and mπ = m, there exists a unique ζ ∈ C such that mζ = m. More precisely, we then have mπj = mζj for all j ∈ Z. Hence each ζ ∈ C is uniquely determined by π. The mapping p : C → C thus is an injection of the set CX of all sets of mutually disjoint cycles = idX into SX . In fact, p is a bijection: For an arbitrary π ∈ SX we choose a set of representatives R for the set of all non-trivial π-orbits in X and deﬁne, for each r ∈ R, ζr to be the cycle with the property rζj = rπj for all j ∈ Z.

Then r∈R ζr = π. We have proved:

1.1.2.For each π ∈ SX there exists a unique set C of mutually disjoint cycles in SX idX such that C = π. The set Cassociated with π is called the cycle decomposition of π. For all n ∈ N we n n have π = ζ∈C ζ , by 1.1.1. This shows, in particular,

1.1.3. If π∈ SX has the cycle decomposition C, then o(π) = lcm{o(ζ)|ζ ∈ C}. σ σ Furthermore, π = ζ∈C ζ for all σ ∈ SX . The ﬁrst part of the following remark shows that each ζσ is again a cycle the nontrivial orbit of which is the image of the nontrivial σ σ σ ζ-orbit under σ. Hence C (:= {ζ |σ ∈ SX }) is the cycle decomposition of π .

4 k i.e., the smallest positive integer such that ζ = idX

7 1.1.4. Let k ∈ N and m1,...,mk be mutually distinct elements of X, σ ∈ SX . Then σ (1) m1 ... mk = m1σ ... mkσ .

(2) m1 ... mk = m 1 m2 m1 m3 m1 mk .

Proof. For all m ∈ X we have (putting mk+1 := m1, ζ := m1 ... mk ) m, if m ∈ {m ,...,m }; mζ = 1 k mj+1, if m = mj for some j ∈ k.

First, this equals m m1 m2 m1 m3 m1 mk and therefore shows (2). Secondly it implies, for any σ ∈ S , X

mζσ = mσ m1σ ... mkσ , hence ζσ = σ m1σ ... mkσ , i. e., (1).

As a consequence, if ζ = m1 ... mk is a cycle of length k in SX , then the conjugacy class ζSX is the set of all cycles of length k. If n ,...,n are mutually distinct elements 1 k of X, the bijection mj → nj (j ∈ k) may be extended to a permutation of X, and any σ such extension σ has the property that ζ = n1 ... nk , by 1.1.4(1). A cycle of length 2 is called a transposition. From 1.1.4(2) and 1.1.2 we obtain the well-known fact that the set of all transpositions is a set of generators of the group SX . Recall that there exists a (unique) homomorphism of SX into {1, −1} such that every transposition is mapped to −1, called the sign homomorphism5. A permutation, written as a product of transpositions, is an element of the kernel of sgn if and only if the number of factors is even, and therefore then also called an even permutation. The subgroup AX := ker sgn of SX is called the alternating group on X and is of index 2 in SX if |X| > 1. It is obviously generated by all products ττ ′ where τ, τ ′ are transpositions. If m, n, n′ are 3 distinct elements of X, then m n m n′ = m n n′ . If m, m′, n, n′ are 4 distinct elements of X, then m n m′ n′ = m m′ n′ m m′ n . Hence a product of two distinct transpositions is either a 3-cycle or a product of two 3-cycles. Summarizing, we obtain

1.1.5. The set of all transpositions is a generating set of SX , the set of all 3-cycles a generating set of AX.

6 Let π ∈ SX , ζ an element of the cycle decomposition C of π, m ∈ X such that mζ = m. If |X| ≥ 4, we choose n ∈ X diﬀerent from mζ−1, m, mζ and put σ := mζ−1 m n . By 1.1.4(1), ζσ and ζ are distinct but not disjoint as mζσ = mσ = n = mζ. It follows that ζσ ∈ C so that πσ = π. This shows:

5 jπ−iπ Assuming that X = n for some n ∈ N, it suﬃces to put π sgn := i

8 σ 1.1.6. If idX = π ∈ SX and |X| ≥ 4, there exists a 3-cycle σ ∈ SX such that π = π. In 7 particular, Z(SX )=  = Z(AX) if |X| ≥ 4.

1 The group A5 is of order 60 (= 2 5!) and the group of smallest order which is simple (i. e., not of order 1 and without nontrivial proper normal subgroups)8 and not of prime order. In fact, An is simple for every n ≥ 5. We need a few preparations before we prove this result. For any subgroup U of SX and m ∈ X we set

StabU (m) := {σ|σ ∈ U, mσ = m}, called the stabilizer of m in U. Obviously, StabU (m) is a subgroup of U, for any m ∈ X.

1.2 Proposition. Let n ∈ N>1, j ∈ n.

π (1) (StabAn (j)) = StabAn (jπ) for all π ∈ Sn,

(2) j∈n StabAn (j) = An if n ≥ 4, 9 (3) StabAn (j) is a maximal subgroup of An and isomorphic to An−1.

Proof. (1) For any π, ρ ∈ Sn we have

π π−1 −1 −1 ρ ∈ (StabAn (j)) ⇔ ρ ∈ (StabAn (j)) ⇔ jπρπ = j and πρπ ∈An

⇔ jπρ = jπ and ρ ∈An ⇔ ρ ∈ StabAn (jπ).

(2) If n ≥ 4, j∈n StabAn (j) contains every 3-cycle. Hence the claim, by 1.1.5. (3) The claim is trivial for n ≤ 3. Let n ≥ 4, X := n {j} and set

f : StabAn (j) → SX , σ → σ|X .

Then f is a monomorphism and leaves the sign unchanged, hence StabAn (j)f ≤AX . If

π ∈AX , we extend π to a permutation σ ∈ StabAn (j) by putting jσ := j. Then σf = π. Hence ∼ ∼ StabAn (j) = StabAn (j)f = AX = An−1 as |X| = n − 1. Now let ϕ ∈ An StabAn (j), i := jϕ. For any k ∈ n {j} we may choose l ∈ n {i, j, k} (as n ≥ 4) and put σ := i k l so that we obtain, by (1),

ϕσ StabAn (k) = StabAn (jϕσ) = (StabAn (j)) ⊆StabAn (j) ∪{ϕ}.

Hence StabAn (j) ∪{ϕ} = An, by (2). The claim follows.

7 Of course, also Z(S3)=  while Z(A3)= A3. 8 A normal subgroup must be a union of conjugacy classes, one of them necessarily being {id5}. The other conjugacy classes have orders 12, 12, 15, 20 so that only the union of all conjugacy classes is a subgroup = , by Lagrange’s theorem. 9A maximal subgroup is a proper subgroup which is not contained in any other proper subgroup, i. e., with respect to the relation ⊆, a maximal element of the set of all proper subgroups.

9 We remark that 1.2 holds likewise if An is replaced by Sn in its formulation.

1.3 Theorem. The group An is simple for every n ∈ N≥5.

Proof by induction on n, the case of n = 5 being taken for granted. Let n ∈ N>5 and ∼ suppose An−1 to be simple. Put B := StabAn (n), and let N An. Then B ∩ N B = An−1, by 1.2(3), hence by our inductive hypothesis either B ∩ N = B (1st case) or B ∩ N =  (2nd case).

In the ﬁrst case we conclude that B < N, as B An by 1.2(1), hence N = An by 1.2(3). Our aim in the second case is to show that N = . Assuming N = , we obtain a contradiction as follows: We have B < BN ≤ An, hence BN = An by 1.2(3). In particular, |N| = |An : B| = n. Let X be the normal subset N  of An. We obtain a homomorphism of An into SX by restricting the usual conjugation within the group An to X: Let f : An → SX , σ → σ¯|X .

Then N is centralized by ker f, hence B ∩ ker f An. Hence B ∩ ker f =  as B is a non-normal simple subgroup of An. It follows that ∼ ∼ ∼ An−1 = B = Bf ≤ SX = Sn−1, hence |SX : Bf| = 2 so that Bf is normal and contains every 3-cycle of SX . By 1.1.5, Bf = AX . Thus any even permutation of the set N which leaves the neutral element ﬁxed is an automorphism of the group N. It is an easy exercise10 to show that this implies n< 5, a contradiction.

The alternating groups An, n ≥ 5, thus form a so-called series of finite simple groups. The most trivial series of this kind is given by the groups Cp of prime order p. A third series of simple groups is given as follows: Let V be an n-dimensional vector space over a field K and det the determinant epimorphism of GL(V ) into the multiplicative group K˙ of K, SL(V ) := ker det, P SL(V ) := SL(V )/Z(SL(V )). Then P SL(V ) is simple unless n =1 or n = 2, |K| ≤ 3. We will give a sketch of a proof but not its details which may be found, e. g., in [H], II, §6. It is remarkable, however, that there is a close analogy between the structure of the proof of the simplicity of An. Therefore we mention the crucial steps and definitions for the case of P SL(V ) together with the corresponding parts for the case of An, which will reveal most similar proof procedures.

“By nature”, the group Sn permutes the “set of points” n. Considering now similarly the set X of all 1-dimensional subspaces of V as a “set of points”, the same statement holds for the group GL(V ). Instead of a mere analogy we have, more precisely, a reduction of the latter phenomenon to the “inborn” action of a symmetric group (on its natural set of points): The elements of GL(V ), originally deﬁned as mappings of V onto V , induce mappings of P(V ) onto P(V ), in particular, of X onto X . In this sense, we may restrict each α ∈ GL(V ) to X . This restriction deﬁnes a homomorphism of GL(V ) into SX . In group theoretic investigations, such “new interpretations” of group elements as 10Assume n ≥ 5. There are distinct elements α, β ∈ X which are not inverses of each other. Put γ := αβ and choose δ ∈ X such that α,β,γ,δ are mutually distinct. By hypothesis, the 3-cycles α β γ , α β δ , α δ β are automorphisms of N. Hence α = βγ, βδ = γ = δα. It follows that δβγ = δα = γ, δβ = id, hence γ = id, a contradiction. 10 permutations on certain suitably chosen sets X play an important role as will be seen on many occasions in the sequel. A transvection of V is a vector space automorphism α of V for which there exists a maximal subspace W of V such that α induces the identity endomorphism on W and on V/W .

In GL(V ) In Sn

SL(V ) := ker det An := ker sgn Let T (V ) be the set of all transvections Let Dn be the set of all 3-cycles on n. of V . T (V ) = SL(V ). Dn = An. T (V ) is a conjugacy class in GL(V ) (for Dn is a conjugacy class in Sn (for n ≥ 5 n ≥ 3 even in SL(V )). even in An). Let X be the set of all 1-dimensional sub- Let j ∈ n. spaces of V , U ∈ X . For any T , T ′ ∈ X {U} there exists For any k,k′ ∈ n {j} there exists ′ ′ α ∈ GL(V ) such that Uα = U, Tα = T . π ∈ Sn such that jπ = j, kπ = k .

{α|α ∈ SL(V ), Uα = U} is a maximal StabAn (j) is a maximal subgroup of An subgroup of SL(V ) if n ≥ 2. if n ≥ 2.

Z(GL(V )) = {µc|c ∈ K˙ } where Z(Sn)= {idn} if n ≥ 3. µc : V → V , v → cv. Theorem Except when n = 2, |K| ≤ 3, 1.3 Theorem An is simple for n ≥ 5. every proper normal subgroup of SL(V ) is contained in Z(GL(V )). Hence P SL(V ) is simple unless n =1 or n =2, |K| ≤ 3.

In a famous paper of 1963, W. Feit and J. G. Thompson [FT] succeeded in proving the long-standing conjecture by Burnside that every non-cyclic finite simple group is of even order. This key result put world-wide efforts of group theorists into motion which, after more than 20 years of work of highest intensity, led in 1983 to a first proclamation (by D. Gorenstein) that a complete classification of all finite simple groups had been achieved. But a readable presentation of the proclaimed classification was completely out of reach, and indeed a number of – even very serious – gaps were discovered in the sequel. More than 25 years of hard work after the first and premature announcement passed until the experts of the area had completed a number of missing steps. A general conviction is generally shared now that the classification is correct.11 But with several thousands of pages of highly specialized lines of reasoning, the classification theorem cannot really be called accessible in its current state. Nobody can exclude that in a complicated theory of this length there might have been overlooked details of underestimated depth. On the contrary, it would be most surprising if this were not the case. The classification consists of a certain number (most authors count 18) of countably-infinite series of finite

11R. A. Wilson describes the state of aﬀairs by the words: “The likelihood of catastrophic errors is much reduced, though not completely eliminated” ([W], 1.4)

11 simple groups and a list of 26 so-called sporadic finite simple groups. Five of the latter, the so-called Mathieu groups, had been discovered already between 1860 and 1870. Not before 1964, a further sporadic group was discovered by Z. Janko (the first Janko group J1 of order 175560). It may be defined as the subgroup of GL(7, 11) generated by the two elements 0100000 −3 2 −1 −1 −3 −1 −3 0010000 −2113133 0001000 −1 −1 −3 −1 −3 −3 2 0000100 , −1 −3 −1 −3 −3 2 −1 .     0000010 −3 −1 −3 −3 2 −1 −1     0000001  1 3 3 −2 1 1 3     1000000  3 3 −2 113 1     The sporadic group of largest order, often called the Fischer-Griess monster group or friendly giant, contains more than 8 1053 elements and arises as the automorphism group of some non-associative commutative algebra of dimension 196882 over the field of 2 elements. The prime divisors of its order are all the primes between 2 and 31, plus 41, 47, 59, 71. It is known to involve all other sporadic simple groups as factor groups of subgroups, with 6 exceptions, the so-called pariahs. One of these exceptions is the group J1. We leave the fascinating topic of the classification project the final version of which is certainly still far from being discovered. The highly specialized techniques developed in its pursuit are not the aim of this course but will become accessible on the grounds of the ideas which are presented here. The interested reader may study the last three chapters of [KS] as an introduction into that area. A description of all finite simple groups (under the assumption that the current classification result is correct) is the ambitious object of [W]. 1.4 Definition. An operating system is a triple (Ω,X,f) where Ω, X are sets and f is a mapping of Ω into XX . The mapping f is called an action of Ω on X, and the elements of Ω are called operators on X (with respect to f). In many cases it may be assumed that the operators are mappings from X into X right from the beginning, i. e., f = id. There are two very different group-theoretic specializations, both of utmost importance. Let G be a group. 1) G is passive: X = G. Let (Ω,G,f) be an operating system where f is a mapping of Ω in End G. Then G is called a group with set of operators Ω. 2) G is active: Ω= G. Let(G,X,f) be an operating system where f is a homomorphism of G into SX . Then f is called a group action of G on X. We will first consider type 2) of an operating system. The group action f is called faithful if f is injective, i. e., if ker f = . If there is no danger of confusion (in particular, if there is a unique group action f considered), we write simply xg instead of x(gf) (where x ∈ X, g ∈ G). Furthermore, for every x ∈ X the set xG := {y|y ∈ X, ∃g ∈ G xg = y}

12 is called the G-orbit of x. The group action is called transitive if xG = X for all x ∈ X. 1.4.1. If G acts on a set X, then the relation ∼ given by G x ∼ y ↔ ∃g ∈ G xg = y (x, y ∈ X) G is an equivalence on X, and the equivalence classes are exactly the G-orbits in X. In particular, the G-orbits form a partition of X.

This partition is usually denoted by X/∼ or X/G. There are two standard actions of G, G both with Ω = G = X:

(a) f : G → SG, g → g¯, i. e., G acts on G via conjugation. Then ker f = Z(G) (see 1.0.1), f is faithful if and only if Z(G) = . Conjugation is written exponentially. Accordingly, the orbit of an element x ∈ G under conjugation, i. e., the conjugacy class of x, is denoted by xG (cf. p. 6).

(b) ρ : G → SG, g → ρg (where ρg : G → G, x → xg), i. e., G acts on G via right multiplication. Then ker ρ = , i. e., ρ is faithful. More precisely, ρ is elementwise ﬁxed-point-free, which means that the set Fix g of all ﬁxed points of g ∈ G is empty unless g =1G. Furthermore, xG = {xg|g ∈ G} = G, i. e., ρ is transitive. As ρ is a monomorphism, we conclude

1.4.2. (Cayley’s theorem) Every group G is isomorphic to a subgroup of SG. Left multiplication does not give a homomorphism, but an anti-homomorphism of G into SG. This implies that the inverse left multiplication λ : G → SG, g → λg−1 (where λh : G → G, x → hx), is a transitive elementwise ﬁxed-point-free action of G on G. We have the following connection between the standard actions of G: 1.4.3. ∀g ∈ G (gλ)(gρ)=¯g =(gρ)(gλ).

1.4.4. If f is a group action of G on a set X und H ≤ G, then f|H is a group action of H on X.

With respect to ρ, λ we have: The H-orbits in G regarding ρ|H (λ|H resp.) are the left cosets xH (the right cosets Hx) where x ∈ G. We write G /.H (G./ H resp.) for the set of all right cosets (left cosets) of H in G. G /.H → G /.H 1.4.5. Let H ≤ G. Then f : G → S , g → is a transitive H G /.H Hx → Hxg x 12 action of G on G /.H, ker fH = x∈G H . −1 x Proof. g ∈ ker fH ⇔ ∀x ∈ G Hxg = Hx ⇔ ∀x ∈ G xgx ∈ H ⇔ ∀x ∈ G g ∈ H x ⇔ g ∈ x∈G H for all g ∈ G. 12The same holds for the inverse left multiplication on G./ H.

13 x The subgroup x∈G H is the largest normal subgroup of G which is contained in H, called the core of H in G and commonly denoted by HG. Recall that any group homomorphism f of G induces a group homomorphism fG/N of a quotient G/N whenever N G such that N ≤ ker f. In particular, this holds for group actions f. Thus if HG ⊇ N G then G/N acts on G /.H via right multiplication. As a ﬁrst application of 1.4.5, we let G be the alternating group and prove the following supplement of 1.2(3): ∼ 1.4.6. Let n ∈ N, H a subgroup of index n of An. Then H = An−1. ∼ Proof. For n ≤ 3 the claim is trivial. If n = 4, we have |H| = 3, hence H = A3. Let n ≥ 5, X := An /.H. By 1.3, HAn =  so that fH is a monomorphism of An into SX . ∼ As  = AX ∩AnfH AnfH = AX, it follows that AnfH = AX . We restrict fH to ′ ′ H and thus obtain an action fH of H on the set X := X {H} because the element ′ ′ H ∈ X is ﬁxed by H. We have ker fH = H ∩ ker fH = . As AnfH = AX , HfH ′ ′ consists of even permutations of X so that fH is a monomorphism of H into AX′ . But 1 ′ ∼ ′ ∼ |H| = n |An| = |AX |, hence H = AX = An−1. Let (Ω,X,f) be an operating system. For all ω ∈ Ω we put

FixX,f (ω) := {x|x ∈ X, x(ωf)= x}, and for all x ∈ X

StabΩ,f (x) := {ω|ω ∈ Ω, x(ωf)= x},

13 in short FixX (ω), StabΩ(x) resp., if there is no doubt about the action f. For all n ∈ N we have a natural “componentwise” action of Ω on Xn, given by

n n n X → X f (n) : Ω → (Xn)X , ω → , (x ,...,x ) → (x (ωf),...,x (ωf)) 1 n 1 n and an action on P(X) by

⊂ P(X) → P(X) f : Ω → P(X)P(X), ω → T → T (ωf) where T (ωf) := {t(ωf)|t ∈ T }. For all T ⊆ X

NΩ,f (T ) := Stab ⊂(T ) = {ω|ω ∈ Ω, T (ωf)= T } Ω,f is called the normalizer,

CΩ,f (T ) := StabΩ,f (x) = {ω|ω ∈ Ω, ∀x ∈ T x(ωf)= x} x∈T the centralizer of T in Ω with respect to f. Again, we write NΩ(T ), CΩ(T ) resp., if the reference to f is obvious.

13 Many authors use the notation CX (ω), CΩ(x) resp. the latter of which is also in concordance with what we deﬁne later for subsets T in place of elements x of X.

14 We now consider again the special situation of an action of a group G: Let f be a ⊂ homomorphism of G into SX for some set X. Then f is a homomorphism of G into SP(X). Since gf is injective for every g ∈ G, we have |T | = |T (gf)| for every T ⊆ X. Setting Pl(X) := {T |T ⊆ X, |T | = l} for every l ∈ N, this implies

⊂ 1.4.7. For all l ∈ N, f induces an action of G on Pl(X).

Obviously, StabG,f (x) is a subgroup of G, for every x ∈ X. Applying this to the action ⊂ f, we conclude that the same holds for NG,f (T ) for every T ⊆ X. If T ⊆ X, f induces an action of NG,f (T ) on T the kernel of which is CG,f (T ). Hence we have

1.4.8. CG,f (T ) NG,f (T ) for all T ⊆ X. From 1.4.1 we obtain ˙ 1.4.9. For all T ⊆ X, T = xxNG,f (T ) where x ranges over a set of representatives of 14 the orbits of NG,f (T ) in T . If there is no reference to some special set X and action f of a group G, then it is tacitly assumed that X = G and the action is the 1st standard action (conjugation), i. e., for any T, U ⊆ G,

g g NU (T )= {g|g ∈ U, T = T }, CU (T )= {g|g ∈ U, ∀x ∈ T x = x}.

g x For arbitrary group elements g, x we have x = x if and only if g = g. Therefore CU (T ) may be interpreted as the set of all g ∈ U ﬁxing T elementwise and likewise as the set of all g ∈ U which are ﬁxed points under the action of T . If T is a subgroup of G, then the permutations of the set T induced by NG(T ) by conjugation are in fact automorphisms of the group T . It follows that NG(T )/CG(T ) is then not only a subgroup of ST but, more restrictively, of Aut T :

1.4.10. If T ≤ G, then NG(T )/CG(T ) is isomorphic to a subgroup of Aut T . 1.5 Deﬁnition. Let Ω be a set acting on sets X, Y . A mapping ϕ of X into Y is then called Ω-compatible if ∀x ∈ X ∀ω ∈ Ω (xω)ϕ =(xϕ)ω. If there exists a bijective Ω-compatible mapping of X onto Y , the X, Y are called Ω- equivalent or Ω-similar, written X ≈ Y . We illustrate these actions considering certain Ω actions of a group G:

1.5.1 Example. Let G act upon a set X via f, x ∈ X, H := StabG(x). If a, b ∈ G such that Ha = Hb, it follows that b = ha for some h ∈ H, hence x(bf)= x(hf)(af)= x(af).

14 ˙ denotes a disjoint union.

15 We put ϕ : G /.H → X, Ha → x(af) (which is well-deﬁned as we have just seen). For all a, g ∈ G we have

((Ha)g)ϕ = x(ag)f = x(af)(gf)=((Ha)ϕ)(gf).

Furthermore, ϕ is injective: (Ha)ϕ = (Hb)ϕ ⇒ x(af) = x(bf) ⇒ ab−1 ∈ H ⇒ Ha = Hb, for every a, b ∈ G. Clearly, ϕ is surjective if and only if the action of G on X is transitive. Hence in this case X and G /.H are G-equivalent. We thus have 1.5.2. Up to G-equivalence, the transitive actions of G are those given by 1.4.5. An action of G on a set X is transitive if and only if X ≈ G /.H for a subgroup H of G. G 1.5.3 Example. Let f be a group action of G on a set X, N G, Y := X/N. Then Y ⊂ is G-invariant (with respect to f) as

(∗) ∀x ∈ X ∀g ∈ G (xN)g =(xgg−1)Ng =(xg)N.

⊂ Therefore, f induces a group action of G on Y , and (∗) shows that

ϕ : X → Y, x → xN, is G-compatible. While ϕ is obviously surjective, it is injective if and only if xN = {x} for all x ∈ X, i. e., if and only if N ⊆ ker f. The hypotheses of 1.5.3 are satisﬁed, for example, if two groups H, M act on the same set X (via fH : H → SX , fM : M → SX ) in such a way that HfH ≤ NSX (MfM ). We put G := NSX (MfM ) and apply 1.5.3 with respect to the normal subgroup MfM of G. Then we have:

1.5.4. Let H, M be groups acting on a set X via group actions fH , fM resp., such that

HfH ≤ NSX (MfM ). Then fH induces an action of H on X/M, and the mapping

ϕ : X → X/M, x → x(MfM ), is H-compatible. We say that the action of M is normalized by the action of H if the hypotheses of 1.5.4 are satisﬁed. This simple concept of linking two group actions, sketched by the following diagram, will turn out to be of fundamental importance.

NSX (MfM )

- M MfM HfH H



16 An even more restrictive condition is that the action of M be centralized by the action of

H, i. e., HfH ≤ CSX (MfM ). A simple example is provided by 1.4.5: Let H ≤ G =: X. The group actions

ρ : G → SG, g → gρg, λ|H : H → SG, , h → λh−1 centralize each other (Gρ, Hλ commute elementwise). Furthermore, G /.H is the set of H-orbits in G. The action on G /.H induced by ρ is given by Hx → Hxg (for any g ∈ G), and ϕ : G → G /.H, x → Hx, is G-compatible. 1.6 Proposition (on transitive group actions). Let G be a group which acts transitively on a set X. Let x ∈ X.

(1) (General Frattini argument) ∀H ⊆ G xH = X ⇔ StabG(x)H = G.

(2) |X| = |G : StabG(x)|. Proof. (1) Let H ⊆ G. If xH = X, g ∈ G, we have xg ∈ xH, hence xg = xh for some −1 h ∈ H, i. e., gh ∈ StabG(x), g ∈ StabG(x)H. Conversely, if StabG(x)H = G, then X = xG = xStabG(x)H = xH. (2) follows from 1.5.1. 1.7 Corollary. Let G be a ﬁnite group acting on a ﬁnite set X, B := X/G.

(1) (class equation of a group action) |X| = B∈B |B|,

(2) ∀B ∈ B ∀x ∈ B |B||StabG(x)| = |G|, G in particular: ∀x ∈ G |x ||CG(x)| = |G|,

1 15 (3) (Cauchy-Frobenius Lemma) |B| = |G| g∈G |FixX g| Proof. (1) Obvious by 1.4.1. (Here the finiteness of G is not needed.) (2) The action of G on an orbit is transitive by definition. Hence the claim follows from 1.6(2). The final assertion is the special case where G acts on G by conjugation. 1 1 (3) We have, by 1.4.1, x∈X |xG| = B∈B x∈B |B| = B∈B 1 = |B|. Hence, applying 1.6(2) in the third step, |G| |Fix g| = 1= |Stab (x)| = = |G||B|. X G |xG| g∈G (g,x)∈G×X x∈X x∈X xg=x

15The equation may be interpreted as follows: The number of orbits equals the “average number of ﬁxed points”.

17 Under appropriate hypotheses on the orders of the involved groups we will now obtain a number of statements of fundamental importance in group theory. Recall that a ﬁnite group the order of which is a power of a prime p is called a p-group. A p′-group is a ﬁnite group the order of which is not divisible by p. These may be viewed as special cases of the notion of a π-group for a set π of prime numbers, which is a group the order of which has only prime divisors in π. The set of all primes ∈ π is denoted by π′.

1.8 Proposition. Let G be a group, p a prime.

(1) If H is a p-subgroup of G, T a ﬁnite H-invariant16 subset of G such that p ∤ |T |, then CT (H) = ∅. (2) Let G be a p-group. If  = N G, then N ∩ Z(G) = . In particular, if G = , then Z(G) = . Moreover, G is soluble.

(3) Let H be a p-subgroup of G. If T is an H-invariant coset of some p′-subgroup of G, then CT (H) = ∅. ′ In particular, if N is a normal p -subgroup of G, then CG/N (H)= NCG(H)/N.

Proof. (1) If CH (T ) = ∅, then the order of each H-orbit in T is a power = 1 of p, by 1.7(2). Hence |T |, being a sum of powers =1 of p by 1.7(1), is divisible by p. (2) For the first assertion we set T := N , H := G in (1). The second assertion is the special case N = G. The third now follows by induction on |G|, applying the inductive hypothesis to G/Z(G) which is a p-group of smaller order than |G| if G = . (3) By hypothesis, |T | is not divisible by p so that the first statement is a special case of (1). Under the hypothesis of the final assertion we may apply this to every coset of N in G and obtain the claim.

1.8.1. If the order of a group G is the square of a prime, then G is abelian.

Proof. By 1.8(2), Z(G) =  so that G/Z(G) is cyclic. The claim follows from 1.0.2.

1.8.2. Let p be a prime and G a p-group, H < G. Then H ⊳ G and |G : H| = p. In max particular, G′ ≤ H.

Proof by induction on |G|. If |G| = 1, there is nothing to prove. Let |G| > 1 and assume the claim holds for p-groups of smaller order. By 1.8(2), Z(G) = , hence there exists a central subgroup C of order p. If C ≤ H, we have H/C < G/C, inductively H/C max is normal and of index p in G/C so that the claim follows. Otherwise H

16with respect to conjugation

18 In these applications of 1.7 we considered the 1st standard action, the conjugation. In the following we shall work with the 2nd standard action, the right multiplication. A useful ﬁrst remark generalizes the well-known assertion that a subgroup of index 2 is always normal: 1.8.3. Let G be a ﬁnite group and H

(k) (2) If B ∈ B , then B = G /.H for some H ∈ Ul(G). (k) (3) |Ul(G)| = |B |.

(4) If l is a power of a prime p, then |Ul(G)| ≡ 1. In particular, Ul(G) = ∅. p The proof of 1.9 will yield in passing the following two elementary number-theoretic remarks: 1.9.1. Let k,l ∈ N. Then kl (1) k| l . (2) If l is a power of a prime p, then 1 kl ≡ 1. k l p Proof of 1.9. (1), (2) Let f be the action ofG on G given by right multiplication, B ∈ B, T ∈ B, GT := NG,f (T ) = Stab ⊂(T ). Since |xGT | = |GT | for all x ∈ G we have, by G,f 1.4.9, |G| |G | |T | = l = . T k Hence k |G| = |B| by 1.6(2).19 If |B|= k we have |G | = |G| = |G| = l = |T | so that |GT | T |B| k x−1 1.4.9 implies T = xGT for an x ∈ T . Now let H := GT . It follows that B = {Tg|g ∈ G} = {xGT g|g ∈ G} = {H(xg)|g ∈ G} = G /.H.

17This shows, in particular, that the ﬁrst claim in 1.8.2 (H ⊳ G) is a consequence of the second (|G : H| = p). 18 Recall that |Ul(G)| = 1 if G is cyclic. – In general it may happen that Ul(G) = ∅, e. g., if l = 6, G = A4. 19 kl By 1.7(1) it follows that k B∈B |B| = |Pl(G)| = l . Clearly, a group of order kl exists (at least the cyclic one) so that we obtain 1.9.1(1).

19 (k) (3) The mapping ψ : Ul(G) → B , H → G /.H is surjective by (2). Being the only element of G /.H which is a subgroup, we recover H from G /.H whence ψ is injective. s s (4) Let l = p for some s ∈ N0. By (1) and 1.7(2) we have k |B| |G| = kp for all B ∈ B. Hence pk |B| unless |B| = k. It follows that kl = |Pl(G)| = |B| + |B| ≡ |B| = k|Ul(G)|, l pk B∈B(k) B∈B B∈B(k) |B|=k hence 1 kl (∗) ≡ |Ul(G)|. k l p As this holds so far for an arbitrary group of order kl, we may apply (∗), in particular, to the cyclic group of order kl for which we know that the right-hand side equals 1. This proves 1.9.1(2). By means of this congruence, (∗) now implies |Ul(G)| ≡ 1. p

1.10 Deﬁnition. Let G be a ﬁnite group, p a prime and t ∈ N0 maximal with the property that pt |G|. a Sylow p-subgroup of G is a subgroup P of G such that |P | = pt. We set t Syl p(G) := {P |P ≤ G, |P | = p }, Syl(G) := Sylp(G). p||G| p prime

Note that  ∈ Syl(G) but Sylp(G)=  for all primes p which do not divide |G|. 1.11 Theorem (Sylow 1872). Let G be a ﬁnite group, p a prime.

(1) |Sylp(G)| ≡ 1, in particular: Sylp(G) = ∅. p

(2) If H is a p-subgroup of G and P ∈ Sylp(G), there exists an element g ∈ G such that H ≤ P g.

(3) Sylp(G) is a class of conjugate subgroups of G, and

|G| ∀P ∈ Sylp(G) |Sylp(G)| = . |NG(P )| Proof. (1) is a special case of 1.9(4). (2) H acts on G /.P via right multiplication. Let B :=(G /.P )/H. By 1.7(1),

p ∤ |G : P | = |B| B∈B and ∀B ∈B |B| |H|, by 1.7(2). Hence there exists an H-orbit of length 1, i. e., there exists an element g ∈ G such that (Pg)H = Pg. It follows that P gH = P g, hence H ≤ P g.

20 g (3) If H, P ∈ Sylp(G), there exists an element g ∈ G such that H ≤ P , by (2). Since g g |H| = |P | = |P |, it follows that H = P . Thus G acts transitively on Sylp(G) by conjugation. By 1.6(1), this implies |Sylp(G)| = |G : NG(P )|. The idea of using the line of reasoning in 1.9(4) to prove 1.11(1) is due to H. Wielandt. Of course, 1.9.1 may be proved independently of this group-theoretic context, and some authors indeed use its assertions as auxiliary statements borrowed from elementary number theory to prove the existence part of Sylow’s theorem following Wielandt. But it is a beneﬁt of the proof that it simultaneously yields the remarks 1.9.1(1),(2), avoiding any separate number-theoretic consideration. It should be emphasized that this is one of many examples where some simple algebraic argument has a number-theoretic consequence – for free. Making use of the cyclic group in the proof is therefore in no way a detail which should be replaced by some other line of reasoning. On the contrary: An analysis of the way in which 1.9.1(2) was obtained may even suggest that Wielandt’s idea can be viewed as an ingenious reduction of the case of an arbitrary ﬁnite group to that of the cyclic group of the same order. Conceptually, Wielandt’s proof of 1.11(1) is extremely simple. Omitting the routine technical steps involved, in the view of a group-theoretic expert it boils down to just three lines (see [W], 1.5). Still it should be noted that 1.9(3) holds without any restriction on the divisor l of |G|.

1.12 Corollary. (Classical Frattini argument) Let G be a group, M a ﬁnite normal subgroup of G, P ∈ Syl(M). Then NG(P )M = G. M G Proof. Let p be the prime such that P ∈ Sylp(M). By 1.11(3), M, N (P ) ﬁnite G M a fortiori G, acts transitively by conjugation on Sylp(M), i. e., P =  P G  Sylp(M) = P . Applying 1.6(1) (where X = Sylp(M), x = P , H =  M), we obtain NG(P )M = G.  By 1.11(1),(3), the index of NG(P ) in G must be both a divisor of |G : P | and congruent 1 modulo p if P is a Sylow p-subgroup of G. There are many examples where these two conditions are extremely restrictive, hence can provide some most important structural information on G. We give a few illustrations.

1.12.1 Example. We determine the groups of order 1225. Let G be a group of order 2 2 1225 = 5 7 , G5 ∈ Syl5(G), G7 ∈ Syl7(G). Then G5 ∩G7 = , hence |G5G7| = |G5||G7| = 2 2 |G| so that G5G7 = G. Now |G : G5| =7 and has the divisors 1, 7, 7 . The only divisor G which is congruent 1 modulo 7 is 1. It follows that NG(G7) = G, i. e., 2 2 G7 G. Similarly, |G : G7| = 5 and has the divisors 1, 5, 5 . The only

G5 G7 divisor which is congruent 1 modulo 5 is 1. It follows that NG(G5) = G, ∼ i. e., G5 G. We conclude that G = G5 × G7. The groups G5, G7 have  orders 25, 49 resp., hence are abelian by 1.8.1. In G5, either there is an element of order 25 (and G5 cyclic in this case) or all non-trivial elements are of order 5

21 ∼ (and G5 = C5 × C5 in this case). The same holds analogously for G7 so that we obtain that G must be, up to isomorphism, one of the (mutually non-isomorphic) abelian groups C52 × C72 , C5 × C5 × C72 , C52 × C7 × C7, C5 × C5 × C7 × C7.

1.12.2. Let G be a group of order 30. Then |Syl3(G)| =1= |Syl5(G)|. Proof. We exploit repeatedly the arithmetic condition mentioned before 1.12.1. The only divisor = 1 of 30 which is congruent 1 modulo p is 10 for p = 3, 6 for p = 5.

Therefore |Syl3(G)|=1 = |Syl5(G)| would imply that G has 20 elements of order 3 and 24 elements of order 5 which is absurd. Hence G has a normal Sylow 3-subgroup G3 or a normal Sylow 5-subgroup G5. But then the 15 elements of G3G5 form a subgroup. Both Sylow subgroups of this subgroup are normal in G3G5 (as the only divisors of 15 are 1, 3, 5, 15). But then NG(G3) and NG(G5) are of order ≥ 15, i. e., of index ≤ 2 in G. This implies that NG(G3)= G = NG(G5), hence the claim. ∼ 1.12.3. Let G be a group of order 60 such that |Syl5(G)| = 1. Then G = A5. In particular, every simple group of order 60 is isomorphic to A5. Proof. We ﬁrst observe that G has no subgroup of index 2: Such a subgroup would have a unique Sylow 5-subgroup, by 1.12.2. The index of its normalizer in G would therefore be ≤ 2, hence = 1 as this index must be ≡ 1 modulo 5. This contradicts our hypothesis. Moreover, G has no normal subgroup of order 2 because its quotient would have a subgroup of order 15, by 1.12.2, i. e., of index 2. But then G would have a subgroup of index 2 which, as we have seen, is not the case.

Now let G5 ∈ Syl5(G). Then 1 ≡ |Syl5(G)| = |G : NG(G5)| 12 so that our hypothesis 5 implies that |Syl5(G)| = 6, |NG(G5)| = 10. The action of G on G /. NG(G5) by right multiplication induces a homomorphism f of G into S6 the kernel of which is a subgroup of NG(G5). We know that | ker f|= 2, and ker f = G5 or ker f = NG(G5) would imply that G5 G which is absurd. It follows that ker f = . Hence f is a monomorphism of G into S6. Since Gf ∩A6 is of index ≤ 2 in Gf and G has no subgroup of index 2, it follows that Gf ≤ A6. Thus Gf is a subgroup of index 6 in A6. It follows that ∼ ∼ G = Gf = A5, by 1.4.6.

22 2 Groups with operators

We shall now consider operating systems of type 1) in 1.4. Natural sets of operators on a group G are 1a) Ω = End G, 1b) Ω = Aut G, 1c) Ω = G where the action f of Ω on G is given by idΩ in 1a), 1b), and by f : G → Aut G, g → g¯ (conjugation) in 1c). An End G-invariant subset of G with respect to conjugation is called fully invariant, an Aut G-invariant subset characteristic, and a G-invariant subset of G is called normal (cf. p. 6) in G. We write H ≤ G to say that H is a characteristic subgroup char of G. It suﬃces to consider an arbitrary non-trivial proper subgroup of G = Cp × Cp (p a prime) to see that normal subgroups need not be characteristic in G. Z(G) is characteristic but generally not fully invariant in G: If G = C2 × S3, then Z(G) is the (unique) direct factor of order 2 of G, and there is an endomorphism of G which maps Z(G) onto a non-central subgroup of order 2 of G. The commutator subgroup G′ = [x, y]|x, y ∈ G 20 of a group G is fully invariant, a fortiori characteristic in G, because [x, y]α = [xα,yα] ∈ G′ for all x, y ∈ G, α ∈ End G. For arbitrary subsets U, V of G we set

[U, V ] := [u, v]|u ∈ U, v ∈ V so that G′ = [G,G]. Recall the following characterization of G′ as the smallest normal subgroup of G with abelian factor group.

2.0.4. Let H ≤ G. Then H G and G/H is abelian if and only if G′ ≤ H.

Every vector space V over a ﬁeld K is an example of an (abelian) group with operator set K, where the action of K on V is given by associating to each scalar c the multiplication by c. More generally, this holds if K is a commutative unitary ring and we have a multiplication between K and an abelian group V satisfying the conditions known from the classical vector space axioms. In this case we call V a K-space. A K-representation of a K-space A is a K-linear mapping of A into the K-space EndKV of all endomorphisms of a K-space V which is then called an A-module over K. In particular, every K-space is a K-module over K. If A is a unitary associative algebra, a representation of A which

20Recall that [x, y] := x−1y−1xy = x−1xy,[x, y] = [y, x]−1. Replacing the conjugation by y by an arbitrary group endomorphism γ, one deﬁnes, more generally, [x, γ] := x−1xγ .

23 is a multiplicative homomorphism of A is called an algebra representation of A and unital if 1A acts as the identity on V . Clearly, every A-module V over K is an example of a group with operators. 2.1 Deﬁnition. Let G be a group and Ω a set of operators on G via a mapping f of Ω into End G. The notation T ⊆ G means that T is an Ω-invariant subset of G, i. e., Ω T ⊆ G and T α ⊆ T for all α ∈ Ωf. If N G, then Ω is also a set of operators of N Ω and of G/N. G is called Ω-simple if G =  and , G are the only Ω-invariant normal subgroups of G. A direct Ω-factor of G is an Ω-invariant normal subgroup N of G for which there exists an Ω-invariant normal subgroup M such that G = N×˙ M 21. The fact that this equation is satisﬁed for some Ω-invariant normal subgroups M, N is expressed by the notation G = N×˙ M. If G =  and G allows only the trivial possibilities that Ω N =  or M =  then G is called directly Ω-indecomposable. A direct Ω-decomposition of G is a set X of Ω-invariant normal subgroups =  of G such that

X = G, ∀N ∈ X (X {N}) ∩ N = . If G = , then ∅ is a direct Ω-decomposition of G. If G= , then {G} is a direct Ω-decomposition of G. If Ω also acts on a further group G∗, then an Ω-compatible group homomorphism of G into G∗ is called an Ω-homomorphism. From this we obtain the notions Ω-monomorphism, Ω-epimorphism, Ω-isomorphism, Ω-endomorphism, Ω-automorphism in the obvious way and write EndΩG (AutΩG resp.) for the set of all Ω-endomorphisms (Ω-automorphisms resp.) of G. We write G ∼= G∗ if there exists an Ω-isomorphism of G onto G∗. One readily Ω veriﬁes the homomorphism theorem for groups with sets of operators (which will be used without further reference): The image of G under an Ω-homomorphism ϕ into G∗ is an Ω-subgroup of G∗, the kernel is a normal Ω-subgroup of G, and Gϕ ∼= G/ ker ϕ. Ω An important example arises with any bipartite Ω-decomposition of G, i. e., a pair π = (U, V ) of (not necessarily normal) Ω-subgroups of G such that UV = G, U ∩ V = . The corresponding projections πU , πV of G onto U, V resp., are deﬁned by the condition that gπU is the unique element u ∈ U such that g ∈ uV , gπV is the unique element v ∈ V such that g ∈ Uv, for every g ∈ G. An Ω-factor of G is a component of a bipartite Ω-decomposition of G. If V is an Ω-factor of G, every U ≤ G such that (U, V ) is an Ω Ω-decomposition of G is called an Ω-complement of V in G.

2.1.1. If π =(U, V ) is a bipartite Ω-decomposition of G, then πU , πV are Ω-compatible mappings. If V G, then πU is an Ω-epimorphism of G onto U and ker πU = V . Proof. If g = uv where u ∈ U, v ∈ V , then gα = uαvα and uα ∈ U, vα ∈ V , for all u˜ α ∈ Ωf. If V G, u, u˜ ∈ U, v, v˜ ∈ V , then uv u˜v˜ = uu˜ v v˜, hence πU is a homomorphism, and clearly surjective.

21i. e., G = NM and N ∩ M =  so that G is isomorphic to the direct product N × M.

24 If (U, V ) is a bipartite Ω-decomposition of G and V G, then (U, V ) is called a semidirect Ω-decomposition of G which is expressed in the form G = U⋉˙ V . The symbol Ω is omitted Ω in a context where no such set plays a role.

2.1.2. Let π := (U, V ) be a semidirect decomposition of G and T G such that πV induces a bijection of T onto V . Then ψ := (U, T ) is a semidirect decomposition of G,

πV |T ψT = idT , ψT |V πV = idV .

Proof. U ∩ T =  as πV |T is injective. For any g ∈ G there exists an element x ∈ T −1 such that gπV = xπV , therefore g, x ∈ Uv for some v ∈ V . It follows that gx ∈ U, g ∈ UT . The two ﬁnal assertions follow from the trivial equivalence v = ut ⇔ t = u−1v (t ∈ T,u ∈ U, v ∈ V ).

2.2 Proposition. Let G be a group, Ω a set of operators on G. Let π := (U, V ) be a semidirect Ω-decomposition of G and T G such that πV induces a bijection of T onto Ω V , ψ :=(U, T ). Suppose T ≤ UG V and set γ : G → G, g → gπU gψT . G −1 γ ψT πU Then γ ∈ AutΩ(G), g g = g for all g ∈ G. In particular, V

T πU γ ψT U [G,γ]= T ≤ Z(UG), [G,γ,γ] ≤ [U, γ]= , V = V = T.  UG Special case: If U G, then G = U×˙ V = U×˙ T , T πU ≤ Z(G). Ω Ω

Proof. We have uγ = u for all u ∈ U, vγ = vψT for all v ∈ V directly from the deﬁnition of γ. In particular, U, T ⊆ Gγ. We observe, for any u, u˜ ∈ U, v ∈ V , t ∈ T ,

(∗) uv =ut ˜ ⇒ ∀x ∈ T xu˜ = xu,

−1 −1 because t =u ˜ uv implies thatu ˜ u ∈ UG ≤ CG(T ) as T ≤ UG V . Now u ∈ uC˜ G(T ) and (∗) follows. To prove that γ is a homomorphism let g,g′ ∈ G, u, u,˜ u′, u˜′ ∈ U, v, v′ ∈ V , t, t′ ∈ T such that uv = g =ut ˜ , u′v′ = g′ =u ˜′t′. Then u′uv′uv = g′g =u ˜′ut˜ ′u˜t. By means of (∗),

(g′g)γ = u′ut′u˜t = u′ut′ut = u′t′ut = g′γgγ.

πU ψT As U ∩ T =  we have g ∈ ker γ if and only if g =1G = g , i. e., g ∈ V ∩ U = . We have shown that γ is an injective endomorphism of G such that U, T ⊆ Gγ . Therefore γ πU is an automorphism and Ω-compatible by 2.1.1. Furthermore, [T , UG] = [T, UG] = . If g ∈ G and u ∈ U, t ∈ T such that g = ut, then g−1gγ = t−1tγ = tπU t = tπU , hence −1 γ ψT πU g g = g . Finally, γV = ψT |V is an isomorphism of V onto T by 2.1.2.

Moreover we observe that, under the hypotheses of 2.2, CG(γ)= U(V ∩ T ).

25 2.3 Deﬁnition. Let G be a group. For any set Y , the set GY of all mappings of Y into G is a group with respect to the (in general non-commutative) operation ∔ deﬁned by

α ∔ β : Y → G, y → (yα)(yβ) for all α, β ∈ GY .

The functiono ˙ : Y →  is its neutral element. Given α ∈ GY , the function Y → G, y → (yα)−1, is the inverse of α. We now consider the special case of Y = G. The set GG is a (multiplicatively written) monoid with respect to the usual composition of mappings, and we trivially have the left-sided distributive law

2.3.1. ∀γ,α,β ∈ GG γ(α ∔ β)= γα ∔ γβ.

Note that End G is a multiplicative submonoid of GG. Clearly,

2.3.2. ∀α, β ∈ GG ∀γ ∈ End G (α ∔ β)γ = αγ ∔ βγ.

Unless G is abelian, End G is not additively closed as idG ∔ idG ∈ End G if and only if G is abelian. The following general criterion for a sum of two endomorphisms to be an endomorphism of G is straightforward:

2.3.3. ∀α, β ∈ End G (α ∔ β ∈ End G ⇔ ∀x, y ∈ G yαxβ = xβyα).

For example, if U, T G such that U ∩ T =  and α, β ∈ End G such that Gα ⊆ U, β G ⊆ T , then α ∔ β ∈ End G. In the special case of 2.2, this holds for α = πU , β = ψT , and we have γ = πU ∔ ψT . An automorphism γ of a group G is called a replacement automorphism (with respect to Ω) if there exist U,T,V G such that the hypotheses Ω 22 of the special case of 2.2 are satisﬁed and γ = πU ∔ ψT . By 2.2 we then also have γ = idG ∔ ψT πU . We write RΩG for the subgroup of Aut G generated by all replacement automorphisms of G. In 2.2 (where U G), the subgroup T πU is the image of an Ω-homomorphism of G/U into Z(U)(≤ Z(G)) as G/U ∼= T . We set Ω

ζ ζ ζ ZG,Ω := {G |ζ ∈ EndΩG, G ≤ Z(U), U =  for some direct Ω-factor U of G}. By 2.2, every replacement automorphism centralizes G/ZG,Ω. Hence we have

2.3.4. For any set of operators Ω on G, [G, RΩG] ≤ ZG,Ω ≤ Z(G). If for all bipartite direct Ω-decomposition (U, V ) of G there is no nontrivial Ω-homomorphism of V into Z(U), then RΩG = .

αi αj If n ∈ N and (α1,...,αn) is an n-tuple of endomorphisms of G such that G , G commute elementwise for any distinct i, j ∈ n, we conclude inductively from 2.3.3 that

j∈n αj is an endomorphism of G and independent of the order of summation. A set Ω of operators of G is called normal if all Ω-subgroups of G are normal in G. If Ω is normal, an Ω-decomposition of G is a set ψ of Ω-invariant subgroups of G such that G is

22The eﬀect of γ is a “replacement” of the direct factor V (in the given Ω-decomposition of G) by T .

26 their (restricted) direct product. If T is one of them and T ∗ := (ψ{T }), then (T ∗, T ) is a special case of a bipartite Ω-decomposition of G. The corresponding projection onto T is an Ω-epimorphism (2.1.1). No confusion will arise if this epimorphism is 23 ∗ denoted by ψT . This is obviously an idempotent element of EndΩG, and ker ψT = T . 24 ψT The projections ψT where T ∈ ψ are pairwise orthogonal , and there images G

(T ∈ ψ) commute elementwise. If ψ is ﬁnite, T ∈ψ ψT = idG. More generally, for any element V of a direct Ω-decomposition π of G we then have, by 2.3.2, ψT πV = T ∈ψ |V ( ψT )πV = idV . T ∈ψ |V Let Ω be a normal set operators of G. An Ω-factor V of G ist called retracting if V =  and ϕ for every idempotent ϕ ∈ EndΩG such that G = V and for every ﬁnite Ω-decomposition ψ of G there exists an element T ∈ ψ such that ψT |V ϕ is an automorphism of V . Clearly, every retracting Ω-factor of G is Ω-indecomposable.

2.4 Lemma. Let G be a group with a normal set of operators Ω, ψ a ﬁnite Ω-decomposition of G into Ω-indecomposable subgroups.

(1) Let (U, V ) be an Ω-decomposition of G, V retracting. Then there exists γ ∈ RΩG such that V γ ∈ ψ, [U, γ]= .

(2) Let π be an Ω-decomposition of G, π0 a subset of π consisting of retracting Ω-factors. α Then there exists α ∈ RΩG such that π0 ⊆ ψ, [ (π π0),α]= .

Proof. (1) Put π :=(U, V ). Let T ∈ ψ such that ψT|V ϕ ∈ Aut V. We claim that

(∗) πV |T is an isomorphism of T onto V. T -V πV ∼ ∼ Put N := V ψT , K := T ∩ker πV . Then V = N ≤ T , T/K = V . Ω Ω Ω V -N ψT Since ψT |V πV ∈ Aut V it follows that N ∩ K = , NK = T , - ˙ ∼ K  hence T = N×K. But T is Ω-indecomposable and N = V = . Ω - Thus K = , N = T , proving (∗).   By 2.1.2, V , T are Ω-complements of U in G. By 2.2, there is a replacement automorphism γ such that V γ = T and [U, γ]= .

(2) For finite sets π0 we prove the claim by induction on |π0|: If π0 = ∅, put α := idG. β Now let V ∈ π0 and assume that there exists β ∈ RΩG such that (π0 {V }) , β [V (π π0), β] = . Put U := (π {V }). Then (U, V ) is an Ω-decomposition of G. Choose γ by means of (1) and set α := βγ. Then α ∈ RΩG and α βγ βγ β γ π0 =(π0{V }) ∪{V } =(π0{V }) ∪{V } ⊆ ψ, [ (ππ0),α] = [ (ππ0),γ]=  Hence the claim holds if π0 is finite. But ψ is finite so that, by what we have proved, π0 contains at most |ψ| elements.

23i. e., it coincides with its square. 24 ′ i. e., ψT ψT ′ = 0 for any two distinct T,T ∈ ψ.

27 If we assume the hypotheses of 2.4 and write πr for the set of all retracting elements of an Ω-decomposition π of G into Ω-indecomposable subgroups (analogously ψr for ψ), α then πr = ψr for some α ∈ RΩG. The case where πr = π is of major importance: 2.5 Theorem (General Krull-Schmidt theorem). Let G be a group with a normal set of operators Ω. Suppose that G has an Ω-decomposition into retracting Ω-subgroups. Then RΩG acts transitively on the set of all ﬁnite Ω-decompositions of G into Ω-indecomposable subgroups.25

In particular, under the hypotheses of 2.5 all members of a ﬁnite Ω-decomposition of G into Ω-indecomposable subgroups are retracting. From 2.3.4 we conclude:

2.6 Corollary. Let G be a group with a normal set of operators Ω. Suppose that for any Ω-factor U of G there is no nontrivial Ω-homomorphism of G/U into Z(U). (In particular, this is the case if there is no nontrivial Ω-homomorphism of G into Z(G).) Then there exists at most one ﬁnite Ω- decomposition of G into retracting subgroups.

Note that the notion of a retracting Ω-factor V is, by deﬁnition, a relative one, depending on (the ﬁnite Ω-decompositions of) the group G and not only on V as a group with operator set Ω. By contrast, the property of being Ω-indecomposable is an absolute one, depending only on the Ω-factor in question, not on G. The following remark, however, shows that certain (internal) structural properties of an Ω-factor imply that it must be retracting – regardless of other Ω-factors of G, in this sense: “absolutely” retracting.

2.6.1. Let G be a group with a normal set of operators Ω, V an Ω-factor =  of G. If 26 every element of EndΩV is either nilpotent or an automorphism, then V is retracting.

ϕ Proof. Let ϕ ∈ EndΩG be an idempotent such that G = V and ψ be a ﬁnite Ω- decomposition of G. Then T ∈ψ ψT |V ϕ = idV . Assume that no summand is an automorphism of V . Then each summand is nilpotent, by hypothesis. From 2.3.3 it follows that every subsum is an Ω-endomorphism of V . Let ψ0 be a minimal subset of ψ such that ψ ϕ is an automorphism α of V . Then |ψ | ≥ 2. Let U ∈ ψ und set T ∈ψ0 T |V 0 0 −1 −1 β := ψU ϕα . The choice of ψ0 implies that β and idV − β = ψT ϕα |V T ∈ψ0{U} |V are nilpotent Ω-endomorphisms of V . They obviously commute, hence n n ∀n ∈ N id =(β ∔ id − β)n = βk(id − β)n−k. V V k V k=0 k n−k But for all k ∈ n ∪{0} we have β = o or (idV − β) = o if n is large enough, a contradiction. We mention important special cases where an Ω-factor turns out to be retracting:

25 ˆ The group RΩG may be replaced by RΩˆ G where Ω is the set of all endomorphisms of G which leave all Ω-factors of G invariant. Note that In G ⊆ RΩˆ G. 26i. e., an element a certain power of which equals o.

28 (A) (Azumaya’s condition) Let K be a commutative unitary ring, A a K-space and W an A-module over K (see p.23). Let V be a non-zero direct A-summand of W such 27 that the ring EndAV is local . Then V is retracting. (B) (Krull-Schmidt chain condition) Let G be a group with a normal set of operators Ω, V be an Ω-indecomposable Ω-factor of G satisfying the ascending and the descending chain condition28 for Ω-subgroups. Then V is retracting.

Clearly, (A) is a special case of 2.6.1. Applying 2.5, we obtain:

2.7 Corollary (Krull-Schmidt-Azumaya theorem). Let K be a commutative unitary ring, A a K-space and W an A-module over K. Suppose that there exists a ﬁnite direct decomposition of W into A-submodules V with the property that EndAV is local. Then AutAW acts transitively on the set of all ﬁnite direct decompositions of W into indecomposable A-submodules.

To prove (B) it suﬃces to show

2.8 Proposition (Fitting’s Lemma). Let V be a group with a normal set of operators Ω, ϕ ∈ EndΩV . Suppose that V satisﬁes the ascending and the descending chain condition for Ω-subgroups. Then there exists a positive integer j such that V = V ϕj × ker ϕj. If V is Ω-indecomposable, then ϕ is either nilpotent or an automorphism.

Proof. The two chains of Ω-subgroups V ≥ V ϕ ≥ V ϕ2 ≥ ,  ≤ ker ϕ ≤ ker ϕ2 ≤ ϕj must terminate after a ﬁnite number of steps. Hence there exists j ∈ N0 such that V = j+1 V ϕ = , ker ϕj = ker ϕj+1 = . Put τ := ϕj. Then V τ /(V τ ∩ker τ) ∼= (V τ )τ = V τ , hence V τ ∩ ker τ =  because otherwise we would obtain a non-terminating ascending chain of Ω-subgroups of V τ . Furthermore, (V τ )τ = V τ implies that for all v ∈ V there exists w ∈ V such that vτ = wτ 2 , hence (w−1)τ v ∈ ker τ. It follows that V τ ker τ = V . If V is Ω-indecomposable, this means that either ker ϕj = V , V ϕj =  or ker ϕj = , V = V ϕj , i. e., ker ϕ = , V = V ϕ. Hence ϕ is nilpotent or an automorphism.

We apply 2.5, observe that RΩ˜ G ≤ AutΩG (with Ω as in footnote 25) and hence obtain 2.9 Corollary (Classical Krull-Schmidt theorem). Let G be a group with a normal set of operators Ω. Suppose that G satisﬁes the ascending and the descending chain condition for Ω-subgroups. Then AutΩG acts transitively on the set of all Ω-decompositions of G into Ω-indecomposable subgroups.

Note that the hypotheses of 2.9 imply also the existence of a ﬁnite Ω-decomposition into Ω-indecomposable subgroups.

27 i. e., the set of non-units of EndAV is additively closed. 28A partially ordered set X satisﬁes the ascending chain condition (descending chain condition resp.) if every chain in X has a greatest (smallest resp.) element. By Zorn’s Lemma, the ascending (descending resp.) chain condition holds in X if and only if every nonempty subset of X contains a maximal (minimal resp.) element. – In our context, we consider the set X of all Ω-subgroups of V which is partially ordered by set-theoretic inclusion.

29 If Ω = In G, an Ω-factor is just a direct factor of the group G. A finite abelian group is the direct product of its nontrivial Sylow subgroups, hence can only be directly indecomposable if it is a p-group for some prime p. Therefore the following result shows that the only directly indecomposable finite abelian groups are the cyclic p-groups. 2.10 Proposition. Let p be a prime and A a finite abelian p-group. (1) If x ∈ A is an element of maximal order, then x is a direct factor of A. (2) A is directly indecomposable if and only if A is cyclic. Proof. (1) We proceed by induction on |A|, the case |A| ≤ p being trivial. Let |A| > p and x ∈ A of maximal order, X := x, Y be the subgroup of order p of X. If X = A there is nothing to prove. Therefore let X < B ≤ A such that |B/X| = p. We claim (∗) B contains a subgroup = Y of order p. By the choice of x, the image of the homomorphism ϕ : B → X, g → gp, does not contain x, hence is a proper subgroup of X. In particular, |B/ ker ϕ| < |X| = |B/Y | so that ker ϕ>Y , implying (∗). B A X p Thus there is a subgroup T = Y of B of order p. It follows that B/T = T x is a cyclic subgroup of A/T of maximal order as o(T x) = o(x) Y and o(T a) ≤ o(a) for all a ∈ A. Inductively there exists a subgroup p T C ≥ T of A such that A/T = B/T ×˙ C/T . It follows that A = X×˙ C. 

(2) If A is cyclic, the subgroups of A form a chain, hence A is directly indecomposable. If A is directly indecomposable, then A is cyclic, by (1). An endomorphism of a finite cyclic p-group A either maps A into its maximal subgroup and hence is nilpotent or is an automorphism. By 2.6.1, it follows that a direct factor which is a finite cyclic p-group is always strongly indecomposable. An arbitrary finite group G trivially has a direct decomposition into directly indecomposable subgroups. ∼ For an abelian p-group G this means that G = Cpk1 ×× Cpkn for some k1,...,kn ∈ N where Cj denotes a cyclic group of order j (j ∈ N0). Without loss of generality we may assume that k1 ≥≥ kn. As the groups Cpi are strongly indecomposable, we obtain the following consequence of 2.5: 2.11 Theorem (Structure theorem for finite abelian p-groups). Let p be a prime, G a finite abelian p-group. Then ∼ G = Cpk1 ×× Cpkn for uniquely determined numbers n ∈ N0, k1,...,kn ∈ N such that k1 ≥≥ kn. The n-tuple (pk1 ,...,pkn ) is called the type of the abelian p-group G.29 If |G| = pm, we have 1 ≤ n ≤ m. G is cyclic if and only if n = 1. In the opposite extremal case, n = m,

29 If m ∈ N and k1,...,kn are positive integers such that k1 ≥≥ kn and k1 + + kn = m, the m n-tuple (k1,...,kn) is called a partition of m. By 2.11, the number of abelian p-groups of order p equals the number of partitions of m.

30 every element =1 G is of order p, k1 = = km = 1; G is then called elementary abelian. Elementary abelian groups play an important role on many occasions in group theory. One of the reasons for this is the following proposition.

2.12 Proposition. Let H be a ﬁnite30 minimal normal subgroup of a group. Then there exists a simple group S such that H =∼ S ×× S for some r ∈ N. r Proof. Let G be the given group, S a minimal normal subgroup of H. For all g ∈ G we have S ∼= Sg H, in particular min

(∗) ∀N H Sg ⊆ N or Sg ∩ N = .

g g We have H ⊇ g∈G S G, hence g∈G S = H. As H is ﬁnite there exists a minimal g (clearly ﬁnite) subset T of G such that g∈T S = H. For every x ∈ T we have g ⊳ x g g g∈T {x} S H, hence S ∩ g∈T {x} S = , by (∗). It follows that {S |g ∈ T } is a direct decomposition of H. Since every normal subgroup of a direct factor of H is normal in H, it follows that S is simple as S was chosen as a minimal normal subgroup of H. Hence the claim where r = |T |. By 2.5, r is unique and S is unique up to isomorphism. Now H is either elementary abelian or a direct product of nonabelian simple groups. In the latter case, H has a unique decomposition into isomorphic simple direct factors (∼= S), by 2.6. Every inner automorphism of G induces a permutation of the set of simple direct factors of H. This action of G is transitive because the product over a G-orbit of simple direct factors of H is a normal subgroup of G, hence equals H by the minimality of H. It follows that there is just one G-orbit, containing all simple direct factors of H. Summing up, we have shown:

2.12.1. Let H be a ﬁnite nonabelian minimal normal subgroup of a group G. Then there exists a unique ﬁnite direct decomposition π of H into isomorphic simple groups. The action of G on π, π → π G → S , g → , π S → Sg is transitive. Its kernel is S∈π NG(S). We now return to the general situation of an arbitrary group G with operator set Ω.

2.13 Deﬁnition. Let G be a group, U ≤ G. A subnormal series from U to G is a ﬁnite set K of subgroups U0,...,Un of G such that U = U0 ⊳ U1 ⊳ ⊳ Un = G. U is called subnormal in G, denoted by U G, if such a series exists.31 For every such chain K we set U i := Ui/Ui−1 for all i ∈ n and K := {U i|i ∈ n}. Furthermore, l(K) := n = |K| is called the length of K. If V G for all V ∈ K, K is called a normal series in G.

30The proof shows that the hypothesis of ﬁniteness of H may be considerably weakened. 31In contrast to normality, subnormality is a transitive relation on the set of subgroups of a group.

31 Let Ω be a set of operators on G. A subnormal Ω-series is a subnormal series K consisting of Ω-subgroups. If U V ≤ G, V/U is Ω-simple and there exists a subnormal Ω-series Ω Ω from V to G, then V/U is called an Ω-composition factor of G. An Ω-composition series of G is a subnormal Ω-series K from  to G such that every element of K is an Ω-composition factor of G. As usual, the preﬁx “Ω-” is dropped if Ω = ∅. In the special case where Ω= G acts by conjugation on G, an Ω-composition series of G is called a chief series of G and an Ω-composition factor is then a chief factor of G.

2.13.1. If a group G satisﬁes the ascending and the descending chain condition for subnormal Ω-subgroups, there exists an Ω-composition series of G.

Proof. We shall make use of the equivalences mentioned in footnote 28. Let X be the set of all subnormal Ω-subgroups V of G such that there exists an Ω-composition series of V . Then X = ∅ as  ∈ X. Let W be a maximal element of X. We claim that W = G: Otherwise W

2.13.2. Let K be a subnormal Ω-series of G, H ≤ G, N G. Then Ω Ω

K∩H := {V ∩ H | V ∈ K} is an Ω-subnormal series of H,

K≡N := {V N/N | V ∈ K} is an Ω-subnormal series of G/N.

If K is an Ω-composition series of G, K≡N is an Ω-composition series of G/N and K∩H is an Ω-composition series of H if H G. Moreover, l(K)= l(K∩N )+ l(K≡N ). Proof. If U ⊳ V for some subnormal Ω-subgroup V of G, then Ω VN G either U ∩H = V ∩H or U ∩H ⊳V ∩H, UN = V N or UN ⊳V N. UN Ω Ω N Suppose that V/U is Ω-simple. Then either V N/UN ∼= V/U or Ω V N = UN. If H is subnormal in G, then a subgroup of H is V subnormal in H if and only if it is subnormal in G. Therefore ∼ either (V ∩ H)/(U ∩ H) = V/U or V ∩ H = U ∩ H. Moreover, U Ω the Ω-simplicity of V/U implies that U(V ∩ N)(= UN ∩ V ) either coincides with V or with U. We have V ∩ N = U ∩ N if and only  if V N = UN. The equation on the lengths follows. The assertions of 2.13.2 with respect to intersections become false if the term “Ω- composition series” is replaced by “chief series” because the intersections of the members of a chief series of G with N in general do not form a chief series of N: There may exist normal subgroups of N between those intersections which are not normal in G. On the other hand, from a chief series of G clearly a chief series of G/N is obtained by factor- izing modulo N. Subnormal Ω-series K, K∗ of G are called similar, denoted by K ∼ K∗, if there exists a

32 bijection β of K onto K∗ such that Uβ ∼= U for all U ∈ K {}. Clearly, ∼ is an equi- Ω valence relation on the set of all subnormal Ω-series of G. Instead of comparing direct decompositions we now compare subnormal series of a group and obtain the following main result:

2.14 Theorem (Jordan-H¨older). Any two Ω-composition series of a group with operator set Ω are similar.

Proof. We proceed by induction on the sum of the lengths of the two Ω-composition series. If one of the two lengths – a fortiori if their sum – equals 0 , the considered group is of order 1 and the claim trivial. For the inductive step, let G be the group with the two Ω-composition series K, K∗ in question, l(K),l(K∗) > 0. Set K˙ := K {G}, K˙ ∗ := K∗ {G}, N := max K˙ , N ∗ := max K˙∗. Then N G, N ∗ G, the quotients G/N, Ω Ω G/N ∗ are Ω-simple, and K˙ , K˙ ∗ are Ω-composition series of N, N ∗ resp. If N = N ∗, then inductively K˙ ∼ K˙ ∗, hence K = K˙ ∪{G} ∼ K˙ ∗ ∪{G} = K∗. If N = N ∗, put D := N ∩ N ∗. Then NN ∗ = G, G/N ∼= N ∗/D, G/N ∗ ∼= N/D. Ω Ω G ˙ ˙ ∗ ˙ ˙ ∗ By 2.13.2, K∩D, K∩D are Ω-composition series of D; K, (K∩D) ∪{N} are ∗ N N ∗ Ω-composition series of N the lengths of which have a sum isomorphisms, of the same chief factors. Furthermore, the same stement holds for all composition series of a group and their composition factors. It should be noted, however, that both the general Krull-Schmidt theorem 2.5 and the Jordan-H¨older theorem 2.14 are uniqueness statements, not existence statements regarding decompositions, composition series resp. If we write P(K) for the direct product of all groups in K for a subnormal series K of G, the Jordan-H¨older theorem may be re-formulated as follows: 2.14’ P(K) =∼ P(K∗) for any Ω-composition series K, K′ of G. Ω Clearly, 2.14 implies 2.14’. Conversely, 2.14’ implies 2.14 by 2.5 as K, K∗ are (up to isomorphism) Ω-decompositions of P(K) into Ω-simple groups.

Finally we observe that the choice of  as the subgroup where the series begins is not necessary in 2.13.1–2.14: With trivial modifications of the proof we obtain the assertion of the Jordan-Hölder theorem more generally with respect to any two not refinable subnormal Ω-series from a given subnormal Ω-subgroup U of G to G.

33 3 Complements

Let G be a group and (U, V ) a semidirect decomposition of G. Then U acts on V by conjugation, and the multiplication in V , the multiplication in U and this action determine the operation of the group G as we have uvu′v′ = uu′vu′ v′ for all u,u′ ∈ U, v, v′ ∈ V . Conversely, this observation gives rise to an important concept of a group construction: 3.1 Deﬁnition. Let U, V be groups and f a group action of U on V . We deﬁne an operation on the cartesian product U × V of the sets U, V by f

′ ∀u,u′ ∈ U ∀v, v′ ∈ V (u, v) (u′, v′) :=(uu′, vu f v′). f

uf ′ u′f (uu′)f ′u′f Obviously, (1U , 1V ) is neutral. The equation (v v ) = v v shows that is f −1 associative, and (u−1, (v−1)u f ) is an inverse of (u, v). Hence U × V is a group with respect to called the semidirect product of V with U with respect to the action f f and denoted by U ⋉ V or V ⋊ U. The natural injections ε : u → (u, 1V ) (u ∈ U), f f ι : v → (1U , v)(v ∈ V ), are monomorphisms (observing that 1U f = idV ). We have

uf uf ∀u ∈ U ∀v ∈ V (1U , v) (u, 1V )=(u, v )=(u, 1V ) (1U , v ), f f implying that (U ε,V ι) is a semidirect decomposition of (U × V, ), and showing that f (vι)uε =(vuf )ι for all v ∈ V , u ∈ U: 3.1.1. The conjugation of V ι by elements of U ε is given by the action f.

If there is no doubt about f, just the symbols ⋉, ⋊ resp., are used32. The direct product of U and V occurs in the case of the trivial action f : U → {idV }. The concept of semidirect product is of fundamental importance in group theory. Frequently semidirect products are used in proofs and are constituents of more complicated group constructions. Therefore it is awkward to carry along throughout the variables ε, ι. For notational simplicity, it is common use to consider the groups U, V as they are as subgroups of their semidirect product, i. e., instead of uε, vι one simply writes u, v resp. A necessary condition to do this is, of course, that the sets U, V have one and only one element in common: their neutral one. It is usually assumed without further comment that this condition is satisﬁed (or that the sets U, V may be arranged this way

32involving the “normal subgroup triangle” in the way that it correctly refers to V .– U ⋉ V = U ε⋉˙ V ι.

34 without problems) when semidirect products occur. Then (U, V ) (instead of (U ε,V ι)) is a semidirect decomposition of U ⋉ V .33 A group G is called a split extension of V f by U if there exists a normal subgroup N and a complement H of N in G such that N =∼ V , H ∼= U. We know that the isomorphism type of a split extension is completely determined by the isomorphism type of the normal subgroup, its complement and the action of the latter on the former by conjugation. Therefore, theorems are important which (under appropriate hypotheses) allow the conclusion that a certain given group is a split extension: With this information frequently a complicated group structure is reduced to less complex (“smaller”) constituents of the group, thus its analysis reduced to less complicated cases. Before we prove certain for this reason important “splitting theorems” we give a number of examples of semidirect products.

Examples.

(1) Let n ∈ N>1. Every subgroup generated by a transposition i j is a complement of the normal subgroup A of S . Hence S ∼= C ⋉ A . For n ≥ 5 it is an easy n n n 2 n consequence of 1.3 that , An, Sn are the only normal subgroups of Sn. Therefore, apart from choosing diﬀerent complements of An, there are no other non-trivial possibilities to write Sn as a semidirect product. It should be noted, however, that for n ≥ 6, the subgroups 1 2 and 1 2 3 4 5 6 , for example, are complements of A which are not conjugate under S (see 1.1.4(1)). In this sense, n n there exist “substantially diﬀerent” complements of An in Sn.

The group S4 has not only the nontrivial normal subgroup A4 but also an elementary abelian normal subgroup of order 4 which has 4 (conjugate) complements in S4, the four point stabilizers which are isomorphic to S3 (cf. 1.2(3) and the comment after ∼ ∼ the proof of 1.2). Thus we also have S4 = S3 ⋉ (C2 × C2). Correspondingly, A4 = C3 ⋉ (C2 × C2). (Here we did not bother to write down the respective group actions f, but we know that these are given by conjugation within Sn after embedding the factors of the abstract semidirect product.)

(2) For every abelian group V the mapping α : V → V , v → v−1, is an automorphism, and is of order 2 if and only if V is not an elementary abelian 2-group. Let f be the homomorphism of C2 into Aut V such that C2f = α. The corresponding semidirect product C2 ⋉ V is called the dihedral group of V . Usually the term refers f

33Frequently this process of sparing the variables ε, ι is called “identiﬁcation” of U,U ε, of V, V ι resp. This at the ﬁrst sight rather mysterious word obtains an exact meaning as follows: If the groups given are arranged in a way that the sets U, V satisfy the necessary condition that U ∩ V = {1U } = {1V }, then ε ∪ ι is an injection of the set U ∪ V into the set U × V and a monomorphism of the group U and of the group V into the group U ⋉ V . Then (by an application of the extension principle, p. 4) f there exists a set W , an operation on W and an extension of ε ∪ ι to an isomorphism of W onto U ⋉ V . Clearly, instead of the latter we may now consider the group W with its subgroups U, V . f This describes the normally tacitly assumed passage to a semidirect product which contains (not only up to isomorphism) the given groups U, V as desired.

35 only to the case where V is cyclic, in particular, if only the order is given and no isomorphism type is mentioned. Thus the notation D2k refers to the dihedral group of order 2k which arises as the semidirect product C2 ⋉ Ck. It is of order 2 if k = 1, f elementary abelian (of order 4) if k = 2, isomorphic to S3 if k = 3, one of the two types of nonabelian group of order 8 if n = 4.34 It is an easy exercise to prove that a group of order 2p (p a prime) must be isomorphic either to C2p or to D2p. (3) If V is a group and U ≤ Aut V , then U acts “by nature” on V , i. e., we may choose f = idU for our group action and obtain a semidirect product of V with U. We know that there are canonical embeddings of V , U into V ⋉ U, and conjugation of an element of [the embedded] V by an [embedded] element α of U is simply obtained by applying the automorphism α. The special semidirect product V ⋉Aut V is called the holomorph of V .

(4) Let f be an action of a group U on a set X, M any group. We are going to show that f induces canonically an action of U on the group (M X , ∔) (see 2.3), thus giving rise ←− to a semidirect product of M X with U. Deﬁning ϕ u to be the composition (u−1f)ϕ for all u ∈ U, ϕ ∈ M X , we read ←−u as a mapping of M X into M X , and we have ←− ←− ←− (∗) ∀u, v ∈ U ∀ϕ ∈ M X (ϕ u ) v =(v−1f)(u−1f)ϕ = ϕuv.

In the following, we shall write xu instead of x(uf) for any x ∈ X, u ∈ U. For ←− ←− ←− all ϕ, ψ ∈ M X , x ∈ X we have x(ϕ ∔ ψ) u = (xu−1ϕ)(xu−1ψ) = x(ϕ u ∔ ψ u ). If ←−u −1 −1 ϕ =o ˙, i. e., xu ϕ = 1M for all x ∈ X, it follows that ϕ =o ˙ as Xu = X. ←−− ←− Given ψ ∈ M X , put ϕ := ψu−1 . Now (∗) ﬁrst implies that ϕ u = ψ whence ←−u is an automorphism of the group M X . Secondly it shows that

U → Aut M X , u → ←−u ,

is a homomorphism. The semidirect product U ⋉ M X based on this action of U on M X is called the wreath product of U and M with respect to f. Its common notation, M ≀ U, alternatively U ≀ M, takes into account the fact that it is canonically given f f by the homomorphism f of U into SX from the very beginning. Explicitly, its operation reads as follows: 35

∀u, v ∈ U ∀ϕ, ψ ∈ M X (u,ϕ) (v, ψ)=(uv, (v−1f)ϕ ∔ ψ) f

Most natural choices for f are the actions ρ, λ of U on itself (X = U) by right multiplication, inverted left multiplication resp.. The wreath products of U and M

34The other one, the so-called quaternion group of order 8 (the multiplicative closure of the 4 standard basis vectors of the quaternion algebra over R), has a unique subgroup of order 2, hence has no nontrivial semidirect decomposition. – D8 has 5 subgroups of order 2. Four of them are non-normal and each of these is a complement of two non-isomorphic normal subgroups of order 4. 35A more general form of the wreath product arises when a group action ˜ of U on M is given and the product ˜ is deﬁned by (u, ϕ)˜(v, ψ) = (uv, (v−1f)ϕ ∔ ψu˜−1) for all u, v ∈ U, ϕ, ψ ∈ M X . f f

36 for these choices are easily seen to be isomorphic and called regular. The regular wreath product of U and M is denoted by M ≀ U or U ≀ M. In the sequel we will prefer the symbol ≀ .

We take a look at the special case where X is ﬁnite, w. l. o. g. X = n where n ∈ N. Then M X = M n, the set of all n-tuples over M.36 The elements of U ≀ M have the form (u, (m1,...,mn)) (where u ∈ U, mj ∈ M) which is the product f (u, (1M ,..., 1M ))(1U , (m1,...,mn)). No confusion will arise if we write this product simply as u(m1,...,mn) – just dropping the 1’s which formally serve to make the distinction between U, M n resp., and their canonical embeddings in U ≀ M. We f know that conjugation of an element of M n by an element of u ∈ U is given by executing ←−u , i. e.,

u ←−u ∀m1,...,mn ∈ M ∀u ∈ U (m1,...,mn) =(m1,...,mn) =(m1u−1 ,...,mnu−1 ).

Thus the i-th component of an n-tuple over M is sent to the (iu)-th position. The complete wreath product arises when U = Sn, f = id. We extend M by one “new” element 0 (∈/ M), setting M0 := M ∪{0}, and deﬁne

n×n Φ: Sn ≀ M → M0 id mj if j = iπ π(m1,...,mn) → (aij) where aij = . 0 otherwise

The matrix associated with π(m1,...,mn) is graphically obtained as follows:

π(m1, ...... , mj, ...... , mn) ↓ ↓ ↓ 0 . For each j, the element  .  . mj is pulled down to  .   .  the i-th place in the j-    0  th column, where i = −1   −1 i = jπ → 0 ... 0 mj 0 ... 0  jπ . All other matrix    0  entries are 0.  .   .     0      Thus the index of the row containing the element mj is determined by π. Con- versely these indices clearly determine π, while the n-tuple (m1,...,mn) is trivially read from the matrix (by “ignoring all zeros”). In particular, Φ is injective. Every matrix in the image of Φ has exactly one entry diﬀerent from 0 in every row and in

36Recall that, by deﬁnition, an n-tuple over M is a function of n into M. The function which maps j to mj ∈ M for all j ∈ n is denoted by (m1,...,mn).

37 37 every column. A matrix with this property is called monomial. We now make M0 into a monoid with zero by extending the operation of M as follows: a0 :=0=: 0a for all a ∈ M0. Moreover, we set a +0 := a =: 0+ a for all a ∈ M0. This allows us to multiply monomial matrices over M0 via the usual matrix multiplication rule. We show that Φ is a homomorphism: Let m1,...,mn,r1,...,rn ∈ M, π, σ ∈ Sn and aij, bij ∈ M0 such that (π(m1,...,mn))Φ = (aij), (σ(r1,...,rn))Φ = (bij). If aikbkj = 0 for some i, j, k ∈ n, it follows that k = iπ, j = iπσ and aik = miπ, bkj = riπσ. Given i, j, this happens for at most one k. Hence

mjσ−1 rj if iπσ = j, ∀i, j ∈ n aikbkj = 0 otherwise. k∈n

As π(m1,...,mn) σ(r1,...,rn) = πσ(m1σ−1 r1,...,mnσ−1 rn), this means that Φ is a homomorphism. Summarizing, the complete wreath product Sn ≀ M is, up to id isomorphism, the group of all monomial n × n matrices over M0. The image of id(m1,...,mn) under Φ is the diagonal matrix diag[m1,...,mn] while the image of π(1M ,..., 1M ) is the permutation matrix (see footnote 37) associated with π. We now prove that the automorphism group of the direct product of ﬁnitely many copies of a simple non-abelian group may be described as a complete wreath product:

3.2 Proposition. Let S be a group, |S|=1 , n ∈ N, G := S × × S. The mappings n

G → G ∆ : (Aut S) × × (Aut S) → Aut G, (α1,...,αn) → α1 α , n (s ,...,s ) → (s ,...,s n ) 1 n 1 n G → G Π: Sn → Aut G, π → , (s ,...,s ) → (s −1 ,...,s −1 ) 1 n 1π nπ are monomorphisms. Let V, S˜n denote the images of ∆, Π resp. Then ∼ (1) Sn ≀ (Aut S) = S˜n⋉˙ V ≤ Aut G id

(2) If S is simple and non-abelian, then S˜n⋉˙ V = Aut G.

Proof. The ﬁrst claim (on ∆) is obvious. For all π, σ ∈ Sn we have πΠ, σΠ ∈ Aut G, and (πσ)Π σΠ πΠ σΠ (s1,...,sn) = (s1σ−1π−1 ,...,snσ−1π−1 )=(s1π−1 ,...,snπ−1 ) = (s1,...,sn) for all s1,...,sn ∈ S which proves that Π is a homomorphism. If π = id, there exists πΠ an i ∈ n such that i < iπ. Then for any s ∈ S  we have (..., 1S,...,s,... ) = i iπ (...... , 1S,... ), hence πΠ = idG. Thus Π is a monomorphism. iπ

α1 αn (1) Let αj ∈ Aut S, π ∈ Sn. If we assume that (s1π−1 ,...,snπ−1 )=(s1 ,...,sn ), i. e., 37A monomial matrix in which every non-zero entry equals 1 is called a permutation matrix. If M = {1}, Φ reduces essentially to an isomorphism of Sn onto the group of all permutation matrices which also explains their name.

38 αj sjπ−1 = sj for all s1,...,sn ∈ S, we have π = id and αj = idS for all j ∈ n. It follows −1 that S˜n ∩ V = . Furthermore, the composition (π Π)((α1,...,αn)∆)(πΠ) maps every α1 αn πΠ α1π−1 αnπ−1 n-tuple (s1,...,sn) over S to (s1π,...,snπ) = (s1 ,...,sn ), which means, in other words, that πΠ ((α1,...,αn)∆) =(α1π−1 ,...,αnπ−1 )∆.

This equation shows that V is normalized by S˜n and, more precisely, that the bijection

Sn ≀ (Aut S) → S˜n⋉˙ V, π(α1,...,αn) → πΠ((α1,...,αn)∆) id is a homomorphism, hence an isomorphism. (2) Let S be simple and non-abelian. By 2.6, G has a unique direct decomposition into ∼ n normal subgroups S1,...,Sn = S. Hence every automorphism γ ∈ Aut G induces a γ permutation πγ of n, deﬁned by the condition that Sjπγ = Sj for all j ∈ n. The mapping

F : Aut G → Sn, γ → πγ ,

γ is a homomorphism. Now γ ∈ ker F if and only if Sj = Sj for all j ∈ n, i. e., if and ∼ ∼ only if γ ∈ V . Hence Sn = SñV/V ≤ (Aut G)/V = (Aut G)F ≤ Sn which implies the claim. Now let G be an arbitrary group and π =(U, V ) a semidirect decomposition of G. We know that the projection πU is a homomorphism of G onto U, but for the other projection πV a different rule holds. The composition of πU with the action of the subgroup U on the normal subgroup V given by by conjugation defines a group action f of G on V . For all u,u′ ∈ U, v, v′ ∈ V we have uvu′v′ = uu′vu′ v′ so that we obtain the following property of πV : hf ∀g, h ∈ G (gh)πV =(gπV ) (hπV ).

Thus πV may be viewed as a “disturbed homomorphism” because the calculation of the value of a product depends on the action of the second factor.

3.3 Definition. Let G be a group which acts on a group M. Write h˜ for the automorphism of M associated with h ∈ G. A crossed homomorphism of G into M is a mapping w : G → M such that ˜ ∀g, h ∈ G (gh)w =(gw)h(hw). Clearly, such a mapping w is a homomorphism if and only if the action of G is trivial on the image of w. Thus crossed homomorphisms are generalized homomorphisms. It is therefore natural that certain properties of homomorphisms hold in a weakened form for a crossed homomorphism w. Let ker w := {g | g ∈ G, gw = 1M }. A first trivial consequence of the definition is that 1G ∈ ker w. We observe further useful properties. 3.3.1. ∀g ∈ G (gw)−1 =(g−1w)g˜,

−1 −1 g˜ as 1M =(g g)w =(g w) (gw) for all g ∈ G.

39 3.3.2. ker w ≤ G.

Proof. We know that 1G ∈ ker w. If g, h ∈ ker w, obviously gh ∈ ker w, and, by 3.3.1, g−1 ∈ ker w. 3.3.3. ∀g, h ∈ G hg−1 ∈ ker w ⇔ gw = hw. g g−1 g−1 Proof. By 3.3.1, (hg−1)w =(hw)g (g−1w)= (hw)(gw)−1 . Hence hg−1 ∈ ker w ⇔ (hw)(gw)−1 =1 ⇔ gw = hw. M 3.3.4. Gw ≤ M ⇔ Gw is G-invariant.

−1 Proof. We have 1M =1Gw ∈ Gw. Hence Gw ≤ M ⇔ ∀g, h ∈ G (gw)(hw) ∈ Gw ⇔ ∀g, h ∈ G (gh)w (hw)−1 ∈ Gw. As(gh)w (hw)−1 =(gw)h˜, the claim follows. 3.3.5. If M ∗ M, then w∗ : G → M/M ∗, g → M ∗(gw), is a crossed homomorphism, G as (gh)w∗ = M ∗(gw)h˜(hw)=(M ∗(gw))h˜M ∗(hw)=(gw∗)h˜ (hw∗) for all g, h ∈ G. We now consider the special case where M = G and G acts by conjugation. The elementary commutator equation [gh, x] = [g, x]h[h, x] (for any g, h, x ∈ G) then provides a ﬁrst and most natural type of example: For every x ∈ G, the mapping

G → G, g → [g, x], is a crossed homomorphism. A major diﬀerence to the general situation is that, given a crossed homomorphism w of G into G, we may apply w again to the elements of Gw. 3.4 Proposition. Let w be a crossed homomorphism of G into G, K := ker w, B := Gw. (1) Bw = B if and only if KB = G,

′ ′ (2) w|B is injective if and only if Kb = Kb for any two distinct b, b ∈ B,

(3) w|B is a permutation of B if and only if B is a right transversal of K in G,

(4) If B G and w|B is a permutation of B, then K is a complement of B in G. Proof. (1) Let g ∈ G. If Bw = B, there exists an element b ∈ B such that gw = bw, hence gb−1 ∈ K by 3.3.3. Thus g ∈ KB. Conversely, let x ∈ K, b ∈ B such that g = xb. Then gw = (xw)b(bw) = bw ∈ Bw. – As for (2), it suﬃces to observe that, by 3.3.3, Kg = Kh for any g, h ∈ G if and only if gw = hw. Combining (1) and (2), we obtain (3). Finally, (4) is just one of the implications in (3) for the case where B G. Given a normal subgroup M of a group G, we will pursue the idea of ﬁnding a complement of M in G in the form of the kernel of some crossed homomorphism of G into M. Under appropriate hypotheses we will see that our crossed homomorphism induces a permutation on M so that our aim will be reached thanks to 3.4(4). Finding a suitable crossed homomorphism, however, wants a special idea and some further preparation. It

40 will, in its general form, provide a powerful group-theoretic instrument to obtain short proofs for quite a number of important results, as will be seen in the sequel.

Let G, M be groups acting on a set X via group actions fG, fM resp., such that

GfG ≤ NSX (MfM ). By 1.5.4, we know that fG induces an action on X/M. A mere re-formulation of this osservation arises when we consider sets of representatives for the M-orbits in X instead of the M-orbits themselves: 1.5.4’ Let G, M be groups acting on a set X such that the action of M is normalized by the action of G. Then the latter induces an action on the set of all sets of representatives of X/M.

If we additionally assume that fM is injective, we obtain a group action of G on M by deﬁning h˜ hfG −1 ∀m ∈ M ∀h ∈ G m := (mfM ) fM .

Now we assume that the hypotheses of 1.5.4’ are satisﬁed and that fM is elementwise ﬁxed-point-free. As long as there is no risk of confusion, we simply write xg instead of x(gfG), xm instead of x(mfM ). If B ∈ X/M, x, y ∈ B, there exists a unique m ∈ M x such that ym = x. This element of M will be denoted by . y 3.4.1. If x, y, z ∈ B, h ∈ G, then

y x −1 (1) = x y z x x (2) = y z y ˜ xh x h (3) = yh y x h˜ Here the first two parts are obvious by the definition, and if m = y , we have (yh)m = yhh−1mh = ymh = xh, proving (3). 3.5 Proposition. Let G, M be groups acting on a set X such that the action of M is elementwise fixed-point-free and normalized by the action of G, Y := X/M. Let R be a set of representatives of Y . For every B ∈ Y let rB the unique element of R ∩ B. Then ˜ r r h r (1) ∀g, h ∈ G Bgh = Bg Bgh , r gh r g r h B B Bg −1 −1 38 (2) ∀g, h ∈ G ϕgh = h ϕg ∔ ϕhg˜ −1 rBg where we define ϕg : Y → M, B → , for all g ∈ G. rBg−1

38This should be read as an equation in the group (M Y , ∔) (see 2.3). If the actions of G and M even centralize each other (i. e.,g ˜ = idM for all g ∈ G), the mapping g → ϕg is a crossed homomorphism of G into (M Y , ∔) with respect to the action of G on M Y as given (for U) in 3.1, Ex. (4).

41 r h r Proof. In (1), both sides equal Bg Bgh , by 3.4.1(2), 3.4.1(3) resp.. (2) follows from rBgh rBgh −1 −1 rBh g −1 g˜−1 (1) and 3.4.1(1) as Bϕgh = =(Bh )ϕg (Bϕh) for all g, h ∈ G, B ∈ Y . rBh−1g−1

3.6 Theorem. Let G be a group, M ≤ G, U := G/MG, g := MGg for all g ∈ G. Then

ω : G → U ≀ M, g → gϕg, ρ¯ is a monomorphism (where ρ¯ is the action of U on G /. M by right multiplication, 1.4.5).

G Proof. We apply 3.5(2) to the case where X = G and G acts by right multiplication ρ, M by inverted left multiplication. Then Y = G /. M and R ⊂ g˜ = idM for all g ∈ G. For all h ∈ G, B ∈ Y we have B(h¯ρ¯)= Bh = B(hρ). U M Hence, by the deﬁnition of the wreath product and by 3.5(2),  MG ←−  h  ∀g, h ∈ G gϕg hϕh = gh(ϕg ∔ ϕh)= ghϕgh.

Thus ω is a homomorphism. If g ∈ ker ω, then g ∈ MG and ϕg =o ˙, hence rB = rBg = rBg for any B ∈ G./ M, implying g =1G.

If M G, then M = MG so that ω in 3.6 is then a monomorphism into the regular wreath product of G/M with M. We conclude:

3.7 Corollary. Let G be a group, M G. Then G is isomorphic to a subgroup of (G/M) ≀ M.

Let M G, U := G/M. The (regular) wreath product U ≀ M is the semidirect product of M U – a group which may be visualized as an “inﬂated” M, a direct product of “many” M’s –, and U. It is certainly a remarkable property of the regular wreath product that this particular split extension contains, up to isomorphism, any group with a normal ∼ ∼ subgroup = M and quotient = U. For example, Cp ≀ Cp contains a cyclic subgroup of order p2, for any prime p. Inductively, it is easily seen that every group of order pn is contained in the iterated wreath product ((Cp ≀ Cp) ≀ Cp) ≀ (n “wreath product factors”). The quaternion group Q8 is contained in C2 ≀ C4 and in C2 ≀ (C2 × C2). Now assume the hypotheses of 3.5 and let X/M be ﬁnite. We write x ∼ y if the elements M x, y ∈ X represent the same orbit of M in X. For any sets R,S of representatives for X/M set S s := M ′ ∈ M/M ′. R r (r,s)∈R×S r ∼ s M S If M is abelian, R may be considered as an element of M. Applying 3.4.1, we obtain

42 3.7.1. If R,S,T are sets of representatives for X/M, h ∈ G, then

S R −1 (I) = R S T R R (II) = S T S ˜ Rh R h (III) = Sh S 3.8 Proposition. Let G, M be groups acting on a set X such that the action of M is elementwise ﬁxed-point-free and normalized by the action of G. Suppose X/M is ﬁnite. For any set of representatives R for X/M put

R w : G → M/M ′, g → . R Rg

′ (1) wR is a crossed homomorphism of G into M/M , R (2) hw = hw [ , h˜] for every h ∈ G and set S of representatives for X/M. S R S Proof. By 3.7.1, we have for all g, h ∈ G

˜ R Rh R R h R (gh)w = = = =(gw )h˜ (hw ), R Rgh Rgh Rh Rg Rh R R ˜ S Rh R S R h R −1 R hw = = = hw = hw [ , h˜]. S Sh Sh Rh R S R S R S

First application of 3.8. Let G be a group, M ≤ H ≤ G, M G, G M abelian with a complement K in H, X := G /.K, n := |G : H| H n finite. Both G and M act via right multiplication on X. The M ab. action of M is elementwise fixed-point-free and normalized by the K action of G. The number of M-orbits in X equals n. Therefore, for  any m ∈ M the definition of wR in this case implies immediately

−n 3.8.1. mwR = m for any set R of representatives for X/M. Let g ∈ G. A coset A = Ka (a ∈ G) has the property that A and Ag represent the same M-orbit if and only if Ag = Am for some m ∈ M, i. e., g ∈ KaM (= Ha). If H G it follows that, with respect to the action of G on X/M, exactly the elements of H have all orbits of length 1.

3.8.2. If H G and Ω is a set of G-endomorphisms of H which normalize K and M, then wR|H is a crossed Ω-homomorphism of H into M.

43 α A A α α Proof. ∀h ∈ H, ∀α ∈ Ω h wR = A∈R Ahα = A∈R Ah =(hwR) . 3.9 Theorem (Gasch¨utz 1952). Let G be a group, M an abelian normal subgroup of G, M ≤ H ≤ G and n := |G : H| ﬁnite. Suppose the mapping M → M, m → mn, is bijective.39 (1) If H splits over M, then G splits over M.

(2) If L, L˜ are complements of M in G and L ∩ H, L˜ ∩ H are conjugate under M, then L, L˜ are conjugate under M. Proof. Choose X as above, with respect to an arbitrary complement of M in H. From 3.8.1 and Proposition 3.4(4) we obtain by our hypothesis on n:

(∗) ker wR is a complement of M in G, for any set R of representatives for the M-orbits in X. This proves (1). Now assume the hypotheses of (2). Then K := L ∩ H is a complement G H of M in H. Choosing R := L /.K as a set of representatives for X/M, M L˜ we see that L ⊆ ker wR as Rg = R for all g ∈ L, hence L = ker wR by L (∗). Clearly we may assume that L˜ ∩H = K. Then the set S := L˜ /.K K  again is a set of representatives for X/M, and L˜ = ker wS. By hypothe- R −n sis, S = m for some m ∈ M. Hence, by 3.8(2), R ∀g ∈ G gw = gw [ ,g]= gw (mw )−1 (mw )g S R S R R R −1 −1 = (mg)wR (m wR)=(mgm )wR,

m m as M is abelian. Therefore, L˜ = ker wS = (ker wR) = L . 3.10 Corollary (Maschke 1897). Let Ω be a commutative unitary ring, G a ﬁnite group, H an ΩG-module, M an ΩG-submodule of H having an Ω-complement in H. If the mapping M → M, m → |G|m is bijective, M has an ΩG-complement in H.

Proof. Let G∗ = G ⋉ H, K an Ω-complement of M in H, R a set of representatives ∗ ∗ for the M-orbits in G /.K. Then |R| = |G : H| = |G|. As H splits over M, ker wR ∗ is a complement of M in G , by (∗) in the proof of Theorem 3.9. By 3.8.2, wR|H is an Ω-homomorphism of the abelian group H into M. Now H ∩ ker wR is an Ω-subspace and normal in G∗, hence an ΩG-complement of M in H. The case where Ω is a field and char Ω ∤ |G| is known as Maschke’s theorem and a ground-laying result in representation theory of finite groups. There exist technically simpler proofs than the one given here, but still it is of some interest to have it as a side result in our context. If the group G in 3.9 is finite, then the mapping M → M, m → mn, is bijective if and

39A group M with this property is called uniquely n-divisible.

44 only if gcd(|M|, n) = 1. A subgroup H with the property that gcd(|H|, |G : H|)=1 is called a Hall subgroup of G,40 and every abelian subgroup of it which is normal in G satisfies the hypotheses in 3.9. In particular, Sylow subgroups are Hall subgroups, so that we have the following typical application of 3.9: 3.11 Corollary. Let G be a finite group, H ∈ Syl(G), M an abelian normal subgroup such that M ≤ H. If M has a complement in H, then M has a complement in G. Complements of M in G are conjugate if and only if their intersections with H are conjugate (under M). Furthermore, the hypothesis in 3.9(1) is trivially satisfied if H = M. For finite groups, this is the basic case of the following more general result: 3.12 Corollary. Let M be a soluble normal Hall subgroup of a finite group G. Then (1) M has a complement in G,

(2) any two complements of M in G are conjugate under M.

′ Proof. Inductively we put, for any group H, H(0) := H, H(j+1) := H(j) for all j ∈ N0. 41 The solubility of M means that there exists a smallest index k ∈ N0 such that M(k) = . We show by induction on k that (1), (2) hold if M(k) = . This is trivial for k = 0. Let k > 0 for the inductive step. M ′ is characteristic in M (cf. p. 23), hence normal in G. By 3.9, there exists a unique conjugacy class of complements of M/M ′ in G/M ′. Given ′ ′ ′ such a complement G1/M (where M ≤ G1 ≤ G), M is a soluble normal Hall subgroup ′ of G1 and M(k−1) = M(k) = . Inductively, there exists a unique conjugacy class of ′ ′ complements of M in G1. Any complement L1 of M in G1 is a complement of M in G, which completes the inductive step for the existence assertion. As for conjugacy, let ′ ′ L2 be any complement of M in G. Put G2 := L2M . Then G2/M is a complement of ′ ′ m m M/M in G/M , hence G2 = G1 for some m ∈ M. It follows that L2 is a complement ′ mm′ ′ ′ of M in G1 so that L2 = L1 for some m ∈ M . It is most remarkable that the assertions of 3.12 remain valid even without the hypothesis that M be soluble although this played such a crucial role in the proof. Regarding the existence assertion (1), we obtain this generalization by a routine induction on |G| : Let

40A fundamental result on soluble groups which, however, will not be proved here, is the following: Theorem (P.Hall 1928, 1937) A finite group G is soluble if and only if for every divisor k of |G| |G| such that gcd(k, k )=1 there exists a subgroup of order k in G. If G is soluble, any two Hall subgroups of the same order are conjugate in G. This theorem may be viewed as a natural generalization of Sylow’s theorem (1.11) in the universe of all soluble finite groups. (The analogue of 1.11(2) for Hall subgroups of soluble finite groups also holds.) Its combination of assertions on (a) existence and (b) conjugacy of subgroups with a specific property in finite soluble groups formed the pattern for various theorems of this type which were discovered in the 1960’s and 1970’s. In these decades, finite soluble groups were intensely studied in this respect by group theorists all over the world, highly influenced by concepts due to W. Gaschütz. 41This number k is called the step number of the soluble group H. It is the number of “steps” the

derived series (H(j))j∈N0 “needs” to reach the trivial subgroup .

45 M be an arbitrary normal Hall subgroup of a ﬁnite group G. If every Sylow subgroup of M is normal in G, M is their direct product, hence soluble by 1.8(2) and we just apply 3.12(1). If there is some non-normal Sylow subgroup P of M, we have NG(P )

46 3.15 Proposition. Let G be a ﬁnite group, A ≤ Aut G such that gcd(|A|, |G|)=1. Then for every prime p there exists an A-invariant Sylow p-subgroup of G.

Proof. Let G∗ := A ⋉ G as in the proof of 3.14, and let P ∈ Syl (G). Since G is a ∗ normal Hall subgroup of G , G ∩ NG∗ (P ) is a normal Hall subgroup of NG∗ (P ), hence ∗ has a complement L in NG∗ (P ), by 3.13(1). By 1.12, NG∗ (P )G = G so that L is a complement of G in G∗, hence must be conjugate to the complement A of G, by 3.13(2). x x x Thus A = L ≤ NG∗ (P ) = NG∗ (P ) for some x ∈ G.

47 4 Transfer

We make a second application of 3.8: Let G be a group, M ≤ G, |G : M| finite, X := G. Consider the action of G by right multiplication and the action of M by inverse ˜ left multiplication. These group actions centralize each other, hence h = idM for all h ∈ G. The M-orbits in G are the right cosets of M in G. For any choice of a right transversal R of M in G we have therefore, by applying both parts of 3.8: ′ 42 4.0.1. wR is a homomorphism of G into M/M and independent of the choice of R. 4.1 Definition. Let G be a group, M ′ ≤ M ∗ ≤ M ≤ G, |G : M| finite, R a right transversal of M in G. Put (M ′m)∗ := M ∗m for all m ∈ M. The transfer of G into M/M ∗ is the homomorphism ∗ ∗ vG→M/M ∗ : G → M/M , g → (gwR) . ∗ ′ (If M = M , the commonly used notation is vG→M instead of vG→M/M ′ .) ∗ ′ As M/M is abelian we will know that G = G if the image GvG→M/M ∗ is non-trivial. The transfer will prove to be an instrument to decide if G has a nontrivial abelian factor group. If G/G′ is finite, G′ is the intersection of the subgroups G[p] := {N|G′ ≤ N ≤ G, G/N p-group} (p prime) so that we may confine ourselves to the question if G = G[p] for a prime p. For an [π] [p] arbitrary set π of primes we put G := p∈π G . 4.1.1. Let G be a finite group, p a prime and ϕ a homomorphism of G into a finite [p] 43 p-group. Then ker ϕ ≥ G , Gϕ = Pϕ for every P ∈ Sylp(G) . In particular, ∼ ∀P ∈ Syl(G) GvG→P = P vG→P , G/ ker vG→P = P/ ker (vG→P )|P ,

42We considered the same combination of group actions in the proof of 3.6. Hence the factors in the product deﬁnition of wR are exactly the quotients as in 3.5(1) for our current choices of group actions. For every g ∈ G, we have in terms of the mappings ϕg −1 −1 ′ rBg ′ −1 −1 (∗) Bϕg M = −1 M = (g wR) = gwR    rB g  B∈G /. M B∈G /. M     as wR is a homomorphism. We may consider the wreath product U ≀ M in 3.6 as a subgroup of ρ¯ SX ≀ M (where X = G /.M) which has a representation as the group of monomial matrices over M0 id (see 3.1(4)). By (∗), gwR is the product of all non-zero entries of the matrix gωΦ (ω as in 3.6, Φ as ′ in 3.1(4), X in place of n), modulo M . We thus obtain an interpretation of gwR as the “signless determinant modulo M ′” of the monomial matrix associated with g. 43More generally, Gϕ = P ϕ holds for every subset P of G such that P G[p] = G.

48 an obvious consequence of the homomorphism theorem.

Given g ∈ G, we may exploit the fact that wR is independent of the choice of R for the calculation of gvG→M/M ∗ , choosing R in dependence of g. This is the main idea behind the following important remark:

4.1.2 (The transfer formula). Let g ∈ G, B be the set of g-orbits in G /. M. For every B ∈ B let xB ∈ G such that MxB ∈ B. Then

−1 ′ |B| xB gvG→M = M g , B∈B

B∈B |B| = |G : M|, and |B| o(g) for all B ∈ B if o(g) is ﬁnite. |B|−1 |B| Proof. If B ∈ B, then B = {Mx B , MxB g,...,MxBg }, MxB g = MxB . By 1.4.1, ˙ |B|−1 B = G /. M so that R := B∈B{xB, xBg,...,xBg } is a right transversal of M in G; in particular, |B| = |G : M|. It follows that B∈B |B|−1 ′ xB xBg xBg ′ xB gwR = M |B| |B|−1 = M |B| , xB g xBg xBg xBg B∈B B∈B −1 |B| by 3.4.1(2). For every B ∈ B, the element m ∈ M with the property m xBg = xB |B| −1 equals xBg xB , which proves the formula. The last assertion is clear by 1.7(2). y ∗ Consequently, if g ∩ M ⊆ M for all y ∈ G, then g ∈ ker vG→M/M ∗ . In the following remark, we consider the image of an element of M under the transfer of G into M/M ∗, choosing M ∗ so that it contains a certain set of commutators:

4.1.3. Let m ∈ M, {m˜ −1m′|m˜ ∈m, ∃x ∈ G m′ =m ˜ x ∈ M} ⊆ M ∗. Then

∗ |G:M| mvG→M/M ∗ = M m .

Proof. By hypothesis,m ˜ x ∈ M ∗m˜ ifm ˜ ∈m and x ∈ G such thatm ˜ x ∈ M. By 4.1.2, −1 ∗ |B| xB ∗ |B| ∗ |G:M| mvG→M/M ∗ = M m = M m = M m . B∈B B∈B 4.1.4. Let G be ﬁnite, F := m−1m′|m, m′ ∈ M, ∃x ∈ G m′ = mx. Then

′ F ≤ M ∩ G ≤ ker (vG→M )|M ≤ ker (vG→M/F )|M .

If gcd(|M/F |, |G : M|)=1, equality holds throughout this subgroup chain.

Proof. The ﬁrst assertion consists of trivial consequences of the deﬁnition of F . By |G:M| 4.1.3, F m = F for all m ∈ M ∩ ker vG→M/F , hence m ∈ F if |M/F |, |G : M| are coprime.

49 4.2 Lemma (Focal subgroup lemma). Let G be a ﬁnite group, p a prime, P ∈ Sylp(G).

−1 ′ ′ ′ x ′ (1) m m |m, m ∈ P, ∃x ∈ G m = m = P ∩ G = ker (vG→P )|P ,

[p] (2) ker vG→P = G .

[p] Proof. (1) follows from 4.1.4 (with M := P ). Furthermore, G ≤ ker vG→P by 4.1.1. Since PG[p] = G and P ∩ G′ = P ∩ G[p], the claim in (2) follows. ′ ∼ [p] The subgroup P ∩G is called the focal subgroup of P . By 4.2, P/ ker (vG→P )|P = G/G . Hence the transfer of G into P , restricted to P , determines if G[p] and G coincide or not. Trivially we have

′ ′ ′ [P, NG(P )] ≤ P ∩ G and P ∩ Q ≤ P ∩ G for all Q ∈ Sylp(G).

An important refinement of 4.2(1) is the result that these subgroups indeed generate the focal subgroup of P . We prepare the proof of this result by an extension of the transfer formula 4.1.2 for the case of an element g of prime power order. By 4.2(1), it suffices to treat this case in order to determine the focal subgroup of P . We will exploit the fact that the subgroups of g then form a chain. The extension of 4.1.2 is based on the observation that any subgroup H of G such that g ∈ H gives rise to a partition of the set of all g-orbits in G /. M (see 4.1.2): Clearly, for every y ∈ G, the H-orbit of My in G /. M splits into g-orbits. Their lengths are divisors of o(g)(= |g|), by 1.7(2). 4.3 Proposition. Let G be a group, M ≤ G, |G : M| finite, g ∈ H ≤ G, o(g) a prime power, B the set of H-orbits in G /. M. For every B ∈ B choose xB ∈ G with MxB ∈ B ′ ′y ∗ such that the g-orbit of MxB is of minimal length. Let M M ∩H |y ∈ G ≤ M ≤ M. Then ∗ |B| x−1 gvG→M/M ∗ = M (g ) B , B∈B |B| x−1 x (g ) B ∈ M, |B| =1 if and only if H ≤ M B .

Proof. For every B ∈ B choose R ⊆ H such that Mx H = ˙ Mx rg, w. l. o. g. B B r∈RB B −1 ∈ R r ∈ R |B| : {j|j ∈ , gj (xB r) ∈ M} 1G B. For every B put r = min N ( ) . This is the length of the g-orbit of Mx r, hence |B| = |B|. We have |B| |B| B r∈RB r 1G r for all r ∈ RB by the choice of xB and the fact that o(g) is a prime power, thus |B| x−1 |B| x−1 |B| x−1 |B| x−1 (g r ) B ∈(g 1G ) B ≤ M. We conclude that (g ) B = (g r ) B ∈ Mand r∈RB

|B| r−1x−1 −|B| x−1 −1 −|B| x−1 ′ x−1 ∗ (g r ) B (g r ) B = [r ,g r ] B ∈ M ∩ H B ≤ M .

Making use of 4.1.2, this implies

−1 −1 −1 ∗ |B|r (xB r) ∗ |B|r x ∗ |B| x gvG→M/M ∗ = M (g ) = M (g ) B = M (g ) B . B∈B r∈R B∈B r∈R B∈B B B

xB xB xB Finally, |B| =1 ⇔ MxB H = MxB ⇔ M H = M ⇔ H ≤ M .

50 ′ |G:M| If H = g, 4.3 reduces to 4.1.2. If H = G we obtain: gvG→M/(M∩G′) = (M ∩ G )g for all g of prime power order, hence for all g ∈ G; this is also a simple consequence of 4.1.2. The proof of 4.4 will make use of the case H = M ∈ Syl(G):

4.4 Theorem (Gr¨un 1935). Let G be a ﬁnite group, p a prime, P ∈ Sylp(G). Then

′ ′ P ∩ G = P ∩ Q |Q ∈ Sylp(G) [P, NG(P )].

∗ ′ Proof. Put P := P ∩ Q |Q ∈ Sylp(G). We show, by induction on o(g), that every ′ ∗ g ∈ P ∩ G is contained in P [P, NG(P )]. This is trivial for g = 1G. For the inductive step let g ∈ P ∩ G′ and o(g) > 1. By 4.3, we have

∗ ∗ |B| x−1 |B| x−1 ′ P = gvG→P/P ∗ = P (g ) B , (g ) B ∈ P ∩ G for all P -orbits B in G /.P. B |B| x−1 |B| x−1 ∗ If |B| > 1, then o((g ) B ) < o(g), hence inductively (g ) B ∈ P [P, NG(P )]. As |B| = 1 if and only if xB ∈ NG(P ), it follows that

−1 −1 |NG(P ):P | −1 xB |B| xB ∗ g ≡ g[g, xB ]= g ≡ (g ) ≡ 1G mod P [P, NG(P )], B∈NG(P )/P |B|=1 B ∗ hence g ∈ P [P, NG(P )] because p ∤ |NG(P ): P |. By 4.4, two different aspects of the embedding of the Sylow subgroup P in G allow to determine the structure of G/G[p]: The intersections of P with the commutator subgroups of all conjugates of P and the largest quotient of P which is centralized by NG(P ). Given any p-group, one can now study the different possible cases which may arise – in dependence of the subgroup structure of P and the p′-subgroups of Aut G – if the given group is to occur as a Sylow p-subgroup of a finite group. For example, in [G], 7.7, there is an analysis of groups with dihedral Sylow 2-subgroups along these lines. The complexity which may be hidden in the two above-mentioned aspects is avoided by brute force if the hypotheses are chosen such that both factors of P ∩G′ in 4.4 are trivial. This is the case in one of the oldest results in the spirit of this chapter, by Burnside (1900): Let p be prime, G a finite group with an abelian Sylow p-subgroup P . Suppose [p] that NG(P ) = CG(P ). Then P is a complement of G in G. On the basis of 4.4, the proof is trivial because we immediately obtain P ∩ G′ = .44 We will give an extended version of Burnside’s result in 5.16. Burnside’s hypothesis is clearly satisfied if p is the smallest prime divisor of |G| and P is cyclic, because each prime divisor = p of |Aut P | then divides p − 1. Thus we observe: 4.5 Corollary. Let G be a finite group with a cyclic Sylow p-subgroup P where p is the smallest prime divisor of |G|. Then P is a complement of G[p] in G.

44We add a direct proof, using only 4.2: Let m ∈ P , x ∈ G such that mx ∈ P . By 4.2(1), it suﬃces −1 to show that mx = m. As m ∈ P ∩ P x and the Sylow p-subgroups of G are abelian, we have − − − x 1 x 1 x 1 z P, P ≤ CG(m). Clearly then P, P ∈ Sylp(CG(m)), hence P = P for some z ∈ CG(m), by −1 1.11(3). It follows that zx ∈ NG(P )= CG(P ), by hypothesis. Hence x ∈ z CG(P ) ⊆ CG(m).

51 Under appropriate hypotheses it may be proved that G and a subgroup G˜ containing a Sylow p-subgroup P of G have isomorphic largest abelian p-factor groups.45 In this direction we observe the following simple consequence of 4.4: ˜ 4.6 Corollary. Let G be a ﬁnite group, p a prime, Z ⊆ P ∈ Sylp(G), G := NG(Z). char ′ ˜ ′ [p] ∼ ˜ ˜[p] Suppose that P ∩ Q ≤ G for all Q ∈ SylpG. Then G/G = G/G .

Proof. We have to show that P ∩ G′ = P ∩ G˜ ′. For the nontrivial inclusion we just ′ have to remark that NG(P ) ≤ G˜, hence [P, NG(P )] ≤ G˜ . The claim then follows from 4.4. By nothing else than making clever use of 1.11, Grün showed that the hypothesis of 4.6 is satisfied with Z = Z(P ) if for all x ∈ G either Z(P )x = Z(P ) or Z(P )x ⊆ P . This result is known as the “2nd theorem by Grün” (see [H], IV, 3.7 or [R], 10.2.3 for a proof which is related to the argument given in footnote 44; furthermore, cf. [KS], 7.1.8). For a 3rd application of 3.8 let G be a group, M ≤ N G, |N : M| finite, X := N. Let M act by inverse left multiplication λ, G by conjugation κ on N. For any g ∈ G, m ∈ M, x ∈ N we have

−1 x(mλ)gκ = m−1xg g =(mg)−1x, hence Mλ is normalized by gκ if and only if Mg ⊆ M. We conclude that the action of NG(M) on N given by conjugation normalizes the action of M on N given by λ. Let R ′ be a right transversal of M in N and wR : NG(M) → M/M the crossed homomorphism from 3.8. We observe

′ −1 |N:M| 4.6.1. ∀m ∈ M mwR =(M m ) mvN→M . Proof. Let m ∈ M. Making use of 3.7.1(II), we have

R R Rm R |N:M| mw = = = = M ′m−1 mv , R Rm (Rm)(mλ) (Rm)(mλ) Rm N→M s −1 because all M-orbits in N /. M are of length 1, and s(mλ) = m for all s ∈ N. 4.7 Lemma. Let G be a group, M ′ ≤ M ∗ ≤ M ≤ N G, N ﬁnite. Suppose that ∗ ∗ ′ ∗ M vN→M ⊆ M /M . For all m ∈ M set m¯ := M m, and deﬁne ψ,ω ∈ End M¯ by −|N:M| ¯ ϕj ¯ ϕj+1 mψ¯ :=m ¯ , mω¯ := mvN→M¯ . Put ϕ := ψ ∔ ω and let j ∈ N such that M = M . ϕj ∗ ∗ Then M¯ is a complemented normal subgroup of (NG(M) ∩ NG(M ))/M .

45A subgroup G˜ with this property is said to control p-transfer of G. For a striking consequence of this see 4.10. Frequently G˜ := NG(P ) controls p-transfer of G, due to the following result: Theorem (Yoshida 1978) Let G be a ﬁnite group, p a prime, P ∈ Sylp(G). Suppose that P has no factor group isomorphic to Cp ≀ Cp. Then NG(P ) controls p-transfer of G. For a proof, see [I], ch. 10.

52 ∗ G Proof. Let D := NG(M) ∩ NG(M ). By 4.6.1 there exists a D crossed homomorphism w of D into M¯ such that mw =mϕ ¯ N for all m ∈ M. From this it is easily seen that wϕi is a crossed M homomorphism of D for all i ∈ N0. By the choice of j we have ¯ ϕj ¯ ϕj+1 j ¯ ϕj j ¯ ϕj ∗ ¯ ϕj M = M ≤ D(wϕ ) ≤ M . Hence D(wϕ ) = M is nor- U : U/M = M j M ∗ malized by D, by 3.3.4, and wϕj induces a permutation on M¯ ϕ . j  By 3.4(4), ker(wϕj)/M ∗ is a complement of M¯ ϕ in D/M ∗.

The hypotheses of 4.7 become considerably simpler if M is contained in ker vN→M¯ . The following corollaries deal with situations in which this is the case: 4.8 Corollary. Let G be a group, M ≤ N G, N finite. Let π be a set of primes which ′ [π] [π] do not divide |N :(M ∩ N )|. Then M/M has a complement in NG(M)/M . ¯ [π] Proof. Let M := M/M . Then vN→M¯ is trivial so that ϕ equals ψ (in the notation of 4.7) and is an automorphism of M¯ . By 4.7 (with j = 1), the claim follows. For every finite group G and prime p we write Op(G) for the smallest normal subgroup of G the quotient of which is a p-group. Clearly, the Sylow p-subgroups of Op(G) are contained in Op(G)′. Thus 4.8 implies: 4.9 Corollary. Let G be a finite group, p a prime, Q ∈ Syl (Op(G)). p G Op(G)

(1) If P ∈ Sylp(G) such that Q ≤ P , there exists a subgroup S of P such P that QS = P , Q ∩ S = Q′. Q Q′ (2) (Gaschütz 1952) If Q is abelian, then Op(G) has a complement in G.  While 4.9(2) is an immediate consequence of 4.9(1), the following application of 4.9(1) is a little less obvious. For a long time, the only known proofs used cohomology or character theory. We thank M. I. Isaacs for friendly calling our attention to the proof in [GI] which is based on transfer methods. A comparison with the approach given here is interesting because the ideas overlap but the strategies are different. ˜ 4.10 Theorem (Tate 1964). Let G be a finite group, p a prime, P ∈ Sylp(G), P ≤ G ≤ G. Then G/G[p] =∼ G/˜ G˜[p] ⇔ G/Op(G) ∼= G/O˜ p(G˜). Proof. The isomorphism on the left is clearly implied by the isomorphism on the right and means that G˜[p] = G˜ ∩ G[p]. We now assume this equation and show that Op(G˜)= G˜ ∩ Op(G). p p ′ G[p] G Set Q := P ∩ O (G), Q˜ = P ∩ O (G˜), F := P ∩ G . Then Op(G) [p] ˜[p] ′ ˜ ˜ ′ F = P ∩ G = P ∩ G , F/Q = (P/Q) , F/Q = (P/Q) . G˜ By 4.9(1), there exists a subgroup S of P such that SQ = P , ′ Op(G˜) S ∩ Q = Q . By 1.8.2, the assumption Q˜ < Q would imply P F QQ˜ ′ < Q, hence QQ˜ ′S

53 5 Nilpotency

The concept of nilpotency makes extensive use of group commutators (see p. 23). There- fore we recall a number of their important properties. The definition and a first vari- ation give, for arbitrary elements x, y of a group G:[x, y] := x−1y−1xy =(y−1x−1yx)−1 = [y, x]−1. For arbitrary subsets U, V of G we conclude that [U, V ] (the subgroup generated by all [x, y] where x ∈ U, y ∈ V ) equals [V, U]. 5.0.1. ∀x, y, z ∈ G [xy, z] = [x, z]y[y, z] = [x, z][[x, z],y][y, z] The first equation in 5.0.1 means that, for every z ∈ G, the mapping [. , z]: G → G, x → [x, z], is a crossed homomorphism. Moreover, it implies

∀x, y, z ∈ G [z,xy] = [xy, z]−1 = [y, z]−1([x, z]−1)y = [z,y][z, x]y = [z,y][z, x][[z, x],y], furthermore 5.0.2. ∀U, V ⊆ G [U, V ] U ∪ V .

5.0.3. [U, V ] ⊆ U ⇔ V ≤ NG(U) for all U, V ≤ G. In particular, [U, G] ⊆ U ⇔ U G. Proof. [U, V ] ⊆ U ⇔ ∀x ∈ U ∀y ∈ V x−1y−1xy ∈ U ⇔ ∀x ∈ U ∀y ∈ V y−1xy ∈ U ⇔ V ⊆ NG(U). The last assertion is the special case V = G. 5.0.4. Let Ω be a set of operators on G, U, V ≤ G. Then [U, V ] ≤ G, Ω Ω because [x, y]α = [xα,yα] for all x, y ∈ G, α ∈ End G. Thus U α ⊆ U, V α ⊆ V implies [U, V ]α ⊆ [U, V ]. 5.0.4 means that [ .,. ] induces an operation on the set of all Ω- subgroups of G. Choosing Ω= G, Ω = Aut G resp., we obtain as special cases

U, V G ⇒ [U, V ] G; U, V ≤ G ⇒ [U, V ] ≤ G. char char Furthermore, 5.0.1 implies 5.0.5. ∀U, V, W G [U, W ][V, W ] = [UV,W ] = [W,UV ] = [W, U][W, V ] 5.0.6. Let U ≤ V ≤ G. Then [V,G] ⊆ U if and only if U, V G and V/U ≤ Z(G/U). Proof. If [V,G] ⊆ U, then [U, G] ⊆ [V,G] ⊆ U ⊆ V , hence U, V G by 5.0.3. Thus [V,G] G by 5.0.4 and v−1vg ∈ [V,G] ⊆ U, hence vU = vgU for all v ∈ V , g ∈ G, i. e., V/U ≤ Z(G/U). Conversely, the latter implies that vU = vgU for all v ∈ V , g ∈ G, hence [V,G] ⊆ U.

54 5.1 Deﬁnition. Let G be a group. A ﬁnite chain K of normal subgroups of G is called G-central if each factor in K (see 2.13) is centralized by G. If U is the smallest, V the largest element of K, then K is called an (ascending) chain from U to V ora(descending) chain from V to U.

1st special case: Put Z0(G) = . Inductively, having deﬁned the normal subgroup Zj−1 of G for some j ∈ N, let Zj(G) be the subgroup of G such that Zj−1(G) ≤ Zj(G) and Zj(G)/Zj−1(G) = Z(G/Zj−1(G)). The set {Zj(G)|j ∈ N0} is called the ascending central series of G. Its union46

Z∞(G) := Zj(G) j∈N0 is called the hypercentre of G. If there exists an n ∈ N0 such that Zn(G) = Zn+1(G), then Zn(G)= Zn+j(G)= Z∞(G) for all j ∈ N. Then {Zj(G)|j ∈ n ∪{0}} is a G-central chain from  to Z∞(G).

2nd special case: Put K0(G) := G. Inductively, having deﬁned the normal subgroup Kj−1(G) of G for some j ∈ N, let Kj(G) := [Kj−1(G),G]. The set {Kj(G)|j ∈ N0} is called the descending central series of G.47 We set

K∞(G) := Kj(G). j∈N0

If there exists an n ∈ N0 such that Kn(G)= Kn+1(G), then Kn(G)= Kn+j(G)= K∞(G) for all j ∈ N. Then, by 5.0.6, {Kj(G)|j ∈ n∪{0}} is a G-central chain from G to K∞(G). The group G is called nilpotent if there exists a G-central chain from  to G. If K is such a chain, then, in particular, every element of K is abelian. Thus we observe: 5.1.1. Every nilpotent group is soluble.

5.1.2. Let n ∈ N, j ∈ n. Then Kj−1(G) ≤ Zn−(j−1)(G) ⇔ Kj(G) ≤ Zn−j(G).

Proof. If Kj−1(G) ≤ Zn−(j−1)(G), then Kj(G) = [Kj−1(G),G] ≤ [Zn−j+1(G),G] ≤ Zn−j(G), by 5.0.6. If Kj(G) ≤ Zn−j(G), then [Kj−1(G),G] ≤ Zn−j(G), hence Kj−1(G) ≤ Zn−j+1(G) by 5.0.6.

Consequently, Kn(G) =  if and only if Zn(G) = G which may be seen by repeated application of 5.1.2:

Kn(G)=  ⇔ Kn(G) ≤ Z0(G) ⇔ Kn−1(G) ≤ Z1(G) ⇔ ⇔ K0(G) ≤ Zn(G).

5.1.3. Let {H0,...,Hn} be a G-central chain such that G = H0 ≥ ≥ Hn. Then Kj(G) ≤ Hj for all j ∈ n ∪{0}.

46 More precisely, the notation should be Zω(G) where ω is the ﬁrst limit ordinal. But we will not dwell on central series in this more general sense here. 47 A traditional notation for its members is obtained by putting γj (G) := Kj−1(G) for all j ∈ N. This shift of subscripts would not be adequate for our present purposes. In other contexts, however, it is reasonable and therefore frequently seen in the literature.

55 Proof. The claim is trivial for j = 0. If j > 0 and Kj−1(G) ≤ Hj−1, then Kj = [Kj−1(G),G] ≤ [Hj−1,G] ≤ Hj, by 5.0.6. 5.1.4. The following conditions are equivalent:

(i) G is nilpotent,

(ii) ∃m ∈ N0 Zm(G)= G,

(iii) ∃n ∈ N0 Kn(G)= . Proof. By 5.1.3, (iii) implies (i), and the converse is trivial. The equivalence of (ii) and (iii) follows from the observed (stronger) consequence of 5.1.2. If G is nilpotent, we conclude the following from what we have seen after 5.1.2:

min{n|n ∈ N0, Zn(G)= G} = min{n|n ∈ N0, Kn(G)= }, i. e., the ascending central chain and the descending central chain of G have the same length. This length is called the nilpotency class of G, denoted by cl(G). From 5.1.3 we conclude

5.1.5. If K is a G-central chain from G to , then cl(G) ≤ l(K).

Nilpotency has certain closure properties which hold trivially but are most useful. Ap- plying 2.13.2 where K is a G-central chain from G to ,Ω= H, = G resp., we obtain

5.1.6. Subgroups and factor groups of nilpotent groups are nilpotent.

From the deﬁnition of the ascending central chain and 5.1.6 we obtain

5.1.7. G is nilpotent if and only if G/Z(G) is nilpotent. 5.1.8. Let X be a set of nilpotent groups. Then × X is nilpotent if and only if there exists an n ∈ N such that cl(G) ≤ n for all G ∈ X , because Zj(× X )= × Zj(G) for all j ∈ N. G∈X 5.1.9. Let Y be a set of normal subgroups of G such that G/N is nilpotent for all N ∈Y, D := Y. If there exists an n ∈ N such that cl(G) ≤ n for all G ∈ Y, then G/D is nilpotent and cl(G/D) ≤ n, because Kn(G) ≤ N for all N ∈Y, hence Kn(G) ≤ D, Kn(G/D)= . 5.1.10. Let Y be the set of all normal subgroups N of G such that G/N is nilpotent. Then Y = K∞(G).

Proof.Clearly K∞(G) ≤ N for all N ∈ Y. On the other hand, K∞(G) = n∈N Kn(G) and Kn(G) ∈Y for all n ∈ N.

56 The group Y in 5.1.10 is called the nilpotent residual of G. If G is ﬁnite, the quotient of Y is nilpotent by 5.1.9. Thus every ﬁnite group has a smallest normal subgroup the quotient of which is nilpotent, namely its nilpotent residual.48 5.2 Proposition. Let G be a nilpotent group.

(1) U

U [U, Zn(G)] ≤ [G,Zn(G)] ≤ Zn−1(G) ≤ U, Zn(G) hence Zn(G) ≤ NG(U) by 5.0.3, U

Let p be a prime and G a p-group. It is easily seen by induction on |G| that G is nilpo- 49 p tent. For an arbitrary finite group G this implies that G/ p prime O (G) is nilpotent, by p 5.1.9. In particular, K∞(G) ≤ p prime O (G). We shall soon see that, in fact, equality holds. 5.3 Main Theorem for finite nilpotent groups. For every finite group G the following are equivalent50: (i) G is nilpotent

(ii) ∀U

(iv) ∀U < G U ⊳ G max (v) ∀P ∈ Syl(G) P G

(vi) ∀P ∈ Syl(G) P ≤ G char (vii) Syl(G) is a direct decomposition of G. Proof. (i)⇒(ii) holds by 5.2(1). (ii)⇒(iii) is proved by induction on |G : U|: The case |G : U| = 1 is trivial. Now let U

48 Note that, despite the name, this term does not refer to the structure of K∞(G), and for inﬁnite groups not even the quotient of Y is necessarily nilpotent. 49For the inductive step, let G = . Then Z(G) =  by 1.8(2). If we assume inductively that the p-group G/Z(G) is nilpotent, it follows that G is nilpotent by 5.1.7. 50A further equivalence will be obtained in 5.12(2).

57 (iii)⇒(iv) is trivial.

(iv)⇒(v): Assume that P G for some P ∈ Syl(G). Then NG(P ) < G so that there exists a maximal subgroup U of G such that NG(P ) ≤ U. By hypothesis, U G. Now 1.12 implies U = NG(P )U = G, a contradiction. (v)⇒(vi) If α ∈ Syl(G), P ∈ Syl(G), then P α ∈ Syl(G). By 1.11(3), there exists an element g ∈ G such that P α = P g. By hypothesis, this means P α = P . (vi)⇒(vii) By hypothesis, Syl(G) is a set of normal subgroups of mutually coprime orders. Hence their product is a direct product and | Syl(G)| = P ∈Syl(G) |P | = |G|, implying that Syl(G)= G. (vii)⇒(i) follows from 5.1.8 as p-groups are nilpotent. p 5.4 Corollary. K∞(G)= p prime O (G) for every finite group G. Proof. Let G be a finite group. We proved one inclusion before 5.3 so that it suffices to p ¯ show K∞(G) ≥ p prime O (G). The quotient G := G/K∞(G) is nilpotent (see 5.1.10), hence the direct product of its Sylow subgroups, by 5.3. It follows that Op(G¯) is the ¯ p ¯ product of all Q ∈ Syl(G) such that p ∤ |Q|, for all primes p. Therefore p prime O (G)= . The claim follows. ∼ 5.5 Corollary. If G is a finite nilpotent group, then Aut G = ×P ∈Syl(G) Aut P .

Proof. Let πP be the projection of G onto P ∈ Syl(G) with respect to the direct decomposition Syl(G) of G (cf. 5.3). Let αP ∈ Aut P for every P ∈ Syl(G). Then α : G → G, πP αP g → P (g ) , is an automorphism of G, and the mapping ∆ : ×P Aut P → Aut G, P αP → α, is a monomorphism. As every P ∈ Syl(G) is characteristic by 5.3, every automorphism α of G normalizes every Sylow subgroup P of G and induces an automorphism αP on P . It follows that ( P αP )∆ = α so that ∆ is an isomorphism. The commutator subgroup of a group G is a particular case of a subgroup of type [U, V ] where U, V ⊆ G. The same holds, more generally, for every member Kn(G) of the descending central series of G if we deﬁne inductively for any sequence (Un)n∈N of subsets of G the left-normed commutator subgroup of (U1,...,Un+1) by

∀n ∈ N [U1,...,Un+1] := [[U1,...,Un], Un+1]. Furthermore, we use the same recursive deﬁnition for elements of G in place of subsets. It is useful to deﬁne, moreover, [U] := U for U ⊆ G,[g] := g for g ∈ G. From 5.0.3 we conclude by a trivial induction

5.5.1. Let Ui ⊆ N G for j mutually distinct i ∈ n. Then [U1,...,Un] ≤ Kj−1(N).

Here, by putting K−1(N)= G, also the case j = 0 is allowed. 5.5.2. Let M, N G, n ∈ N. Then n

[MN,..., MN]= [U1,...,Un] ≤ (Kj−1(M) ∩ Kn−j−1(N)). n n j=0 (U1,...,Un)∈{M,N}

58 Proof. The claimed equality is clear for n = 1. Inductively we conclude, using 5.0.5,

[MN,..., MN] = [W, MN] = [W, M][W, N]= n+1 [U1,...,Un+1] where n+1 (U1,...,Un+1)∈{M,N}

W = n [U ,...,U ]. The claimed inequality follows from 5.5.1. (U1,...,Un)∈{M,N} 1 n 5.6 Corollary (Fitting 1938). Let M, N be nilpotent normal subgroups of a group G. Then MN is a nilpotent normal subgroup of G, and cl(MN) ≤ cl(M)+ cl(N).

Proof. Put n := cl(M)+ cl(N) + 1. Then Kj−1(N)=  or Kn−j−1(M)=  for all j such that 0 ≤ j ≤ n. Thus 5.5.2 implies Kn(MN)= . The claim follows. 5.7 Deﬁnition. Let G be a group. Then

F (G) := N NEG N nilpotent is called the Fitting subgroup of G. The Fitting subgroup may be viewed as a dualization of the nilpotent residual, where intersections of normal subgroups with nilpotent quotient are replaced by products of nilpotent normal subgroups. Clearly,

5.7.1. F (G) ≤ G. char

5.7.2. Z∞(G) ≤ F (G), because for every n ∈ N, Zn(G) is a nilpotent normal subgroup of G: {Zj(G)|j ∈ n} is a ﬁnite G-central chain, a fortiori a ﬁnite Zn(G)-central chain from  to Zn(G). It follows that Zn(G) ≤ F (G), hence the claim.

The tiny example G = S3 shows already that, in general, Z∞(G) is properly contained in F (G). We have Z∞(S3)= < A3 = F (S3).

Without any restriction on G, the hypercentre Z∞(G), a fortiori F (G), need not be nilpotent, as may be seen from 5.1.8. But 5.6 is generalized by a simple induction to

5.7.3. The product of ﬁnitely many nilpotent normal subgroups of G is nilpotent. In particular, F (G) is nilpotent if G is ﬁnite.

Hence every ﬁnite group has a largest nilpotent normal subgroup, namely, its Fitting subgroup.

5.7.4. Let G be ﬁnite. Then ∀N G F (G) ∩ N = F (N).

Proof. As F (N) ≤ N G, F (N) is a nilpotent normal subgroup of G. Furthermore, char F (G) ∩ N is a nilpotent normal subgroup of N. The claim follows.

5.7.5. Let G be a group, p a prime, M, N G, M,N p-groups. Then MN G and MN is a p-group,

59 |M||N| because |MN| = |M∩N| is a power of p. We set Op(G) := N. NEG N p-group

As every p-group is nilpotent, we have p prime Op(G) ≤ F (G).

5.7.6. Let G be ﬁnite. Then p prime Op(G)= F (G).

Proof. F (G) is nilpotent by 5.7.3. For all primes p, Op(F (G)) is its (by 5.3 unique)

Sylow p-subgroup. Hence Op(F (G)) ≤ Op(G), F (G)= p Op(F (G)) ≤ p Op(G).

It is not diﬃcult to see that the left-normed commutators [x1,...,xn+1](xi ∈ G) generate the group Kn(G), for every n ∈ N0. As a consequence we have the following equivalence for every n ∈ N0:

G is nilpotent and cl(G) ≤ n ⇔ ∀x1,...,xn+1 ∈ G [x1,...,xn+1]=1G.

It is remarkable that the property of being nilpotent is for ﬁnite groups already implied by a condition of this kind in which only two variables for group elements occur; this was discovered by M. Zorn. There is a strong analogy with a basic result in the theory of Lie algebras from where also the terminology (of “Engel elements”) has been adopted: An element g of a group G is called 51

right Engel if: ∀x ∈ G ∃n ∈ N [g,x...,x]=1G, n

left Engel if: ∀x ∈ G ∃n ∈ N [x,g,...,g ]=1G. n A right Engel element is thus an element which is “treated nilpotently” by all mappings [ . , x] (x ∈ G) while a left Engel element is characterized by the property that the mapping [ .,g] is nil, i. e., “acts nilpotently” on every x ∈ G. An Engel element is a group element which is right or left Engel. The main result about Engel elements for ﬁnite52 groups is the following:

Theorem (Baer 1957) Let G be a ﬁnite group, g ∈ G.

(1) g is right Engel if and only if g ∈ Z∞(G), (2) g is left Engel if and only if g ∈ F (G).

51Huppert exchanges the meaning of a left and a right Engel element in his deﬁnition ([H], §6, 6.12). 52 More generally, Baer [B] considers group elements the normal subgroup closure of which is noetherian, i. e., all of its subgroups are ﬁnitely generated. In particular, his results hold for noetherian groups.

60 Consequently, every right Engel element of a finite53 group is left Engel, by 5.7.2. It is still unknown if this statement is true for arbitrary groups.54 A main step in the proof of Baer’s theorem is to show that a finite group is nilpotent if it has a set of generators which are Engel elements. A nontrivial part consists in the preliminary result that such a group must be soluble. A detailed proof may be found in [H], III, §6. (But be aware of footnote 51!) Engel conditions in infinite groups have been intensely, but still not conclusively discussed in numerous papers. The Fitting subgroup has a major impact on the study of the structure of a finite soluble group because of the following result:

5.8 Lemma. Let G be a ﬁnite soluble group. Then CG(F (G)) ≤ F (G).

Proof. Let C := CG(F (G)), D := F (G) ∩ C(= Z(F (G))). By 1.4.8, G C G. Being contained in F (G), D is centralized by C, hence F (G) D ≤ Z(C). Assume that C = D. Then let D < M ≤ C such that M/D G/D. As G/D is soluble, M/D is elementary abelian M C min D (2.12). Hence Z2(M)= M so that M is nilpotent. Thus M ≤ F (G), a contradiction. Therefore C = D ≤ F (G).

For any normal subgroup N of a group G, G/CG(N) is isomorphic to a subgroup of Aut G, by 1.4.10, and Z(N)= N ∩ CG(N). In the special case of the Fitting subgroup of a ﬁnite group G we obtain, more speciﬁcally, by 5.5 and 5.8:

5.8.1. Let G be a ﬁnite group. Then G/CG(F (G)) is isomorphic to a subgroup of

P Aut P where P ranges over Syl(F (G)). If G is soluble, CG(F (G)) = Z(F (G)).

Consequently, if G is a ﬁnite soluble group and P1,...,Pr are the Sylow subgroups =  of F (G), D := Z(F (G)), then

• D is an abelian normal subgroup of G isomorphic to Z(P1) ×× Z(Pr),

• G/D is isomorphic to a subgroup of Aut P1 ×× Aut Pr. Thus every finite soluble group G, viewed as an extension of D by G/D, is composed by constituents which may be structurally derived from p-groups, namely from the Sylow subgroups of F (G). For a given finite set of p-groups we have obtained a “structural bound” for the possible finite soluble groups having their direct product as its Fitting subgroup. More generally, such a structural restriction holds for all groups G with the property that CG(F (G)) ≤ F (G). For this reason, groups with this property are called constrained.

53more generally: noetherian, see footnote 52 −1 −1 54The transformation [x, g, . . . ,g] = [g−1,gx ,...,gx ]xg (which is a simple consequence of 5.0.1) n+1 n shows that the inverse of a right Engel element h is always left Engel. Note that the hypothesis is not fully used here as it suﬃces that h(= g−1) be treated nilpotently by all conjugates of h−1.

61 5.9 Deﬁnition. Let G be a group. Then

Φ(G) := H H < G max orH=G

′ 55 is called the Frattini subgroup of G. For example, Φ(D8)= Z(D8)= D8 is of order 2, Φ(S3) = , Φ(Cpn ) is the unique maximal subgroup of Cpn , Φ(Q, +) = Q as the group (Q, +) has no maximal subgroup.56

p 5.9.1. Let p be a prime and G abelian such that g =1G for all g ∈ G. Then Φ(G)= .

Proof. For every g ∈ G choose (by Zorn’s Lemma) U ≤ G maximal with g∩U = . Since o(g)= p it follows that G = U ∪{g}, U < G. Hence g∈ / Φ(G). max 5.9.2. Φ(G) ≤ G, char because H < G ⇔ Hα < G for all α ∈ Aut G. max max 5.9.3. Let U, V ≤ G such that G = U×˙ V . Then Φ(G) ≤ Φ(U)Φ(V ).

Proof. The mapping U → G/V , u → Vu, is an isomorphism. Hence

Φ(G) ≤ H = Φ(U)V, V ≤H < G max orH=G likewise Φ(G) ≤ Φ(V )U. It follows that Φ(G) ≤ Φ(U)V ∩ UΦ(V )=Φ(U)Φ(V ), because every element of G has a unique representation uv with u ∈ U, v ∈ V . The following remark is trivial:

5.9.4. Let U ≤ H for some H < G. Then G = Φ(G)U. max We call G weakly noetherian if every proper subgroup of G is contained in a maximal subgroup of G.

5.9.5. Let G be weakly noetherian, T ⊆ Φ(G). Then we have

(∗) ∀X ⊆ G (X = G ⇒X T = G)

Proof. Let X ⊆ G such that X T = G. By hypothesis, X T ≤ H for some H < G. Then X ⊆ (X T ) ∪ T ⊆ H as T ⊆ Φ(G). Hence X ≤ H

55Let p be a prime. A p-group G is called extra-special if G is elementary abelian or Φ(G)= Z(G)= G′ is of order p. From 1.0.2 and 1.8(2) it follows easily that every non-abelian group of order p3 is extra-special. 56A maximal subgroup of an abelian group has prime index. But nQ = Q for all n ∈ N. Hence (Q, +) has no proper subgroup of ﬁnite index.

62 The converse of 5.9.5 holds without any hypothesis about G: Let T ⊆ G with the property (∗) and assume that T ⊆ H for some maximal subgroup H of G. Putting X := H ∪ T , we then have G = X, hence, by (∗), G = X T ≤ H, a contradiction. Therefore, if G is weakly noetherian, the subsets of Φ(G) are exactly those subsets of G which can be dispensed with in any set of generators of G. In particular, this characterization holds in every finite group. Furthermore, it should be noted that the assertion of 5.9.5 holds for an arbitrary group G if T is finite. It suffices to prove this for the case |T | = 1 as the claim for arbitrary finite T then follows by routine induction. Let X be a set of generators of G and g the unique element of T . If we assume X {g}= G, we find by Zorn’s Lemma a subgroup H which is maximal subject to the conditions that X {g} ⊆ H, g ∈ H. For all y ∈ G H the subgroup H ∪{y} then contains g, hence X, and therefore equals G. Thus H < G. But then g ∈ Φ(G) ≤ H, a contradiction. For this reason the elements max of Φ(G) are also called the non-generators of G.

5.9.6. Let G be weakly noetherian, X ⊆ G. Then X = G ⇔ {Φ(G)x|x ∈ X} = G/Φ(G).

Proof. Clearly, the cosets of the elements of a set of generators modulo a normal subgroup generate its factor group. If Φ(G)x|x ∈ X = G/Φ(G), then Φ(G) ∪ X = G, hence X = G by 5.9.5.

5.9.7. Let N G, N weakly noetherian. Then Φ(N) ≤ Φ(G).

Proof. Let H < G and assume Φ(N) ≤ H. From 5.9.2 we conclude Φ(N)H = G, hence max N = Φ(N)(H ∩N). Then, by 5.9.4, H ∩N = N so that Φ(N) ≤ H, a contradiction. Clearly, direct factors of weakly noetherian groups are weakly noetherian. Therefore we conclude from 5.9.3 and 5.9.7 that G = U×˙ V implies Φ(G)=Φ(U)Φ(V ) if G is weakly noetherian.

We shall need the following simple remark:

5.9.8. Let A be an abelian normal subgroup of G and S ≤ G such that SA = G. Then S ∩ A G.

Proof. S ∩ A S because A G, and S ∩ A A because A is abelian.

5.10 Proposition. Let G be a group.

(1) Let Φ(G) ≤ N G. If N/Φ(G) is nilpotent, then N/Φ(G) is abelian.

(2) An abelian minimal normal subgroup A of G has a complement in G if and only if A ≤ Φ(G).

63 Proof. (1) Without loss of generality we may assume that Φ(G) = , N is nilpotent. We have to show that N ′ =  and assume that N ′ = . Then there exists a maximal subgroup H of G such that N ′ ≤ H, hence HN ′ = G. Let D := H ∩ N. Then there exists a smallest j ∈ N such that DZj(N)= N. Put M := Zj−1(N). Then D ≤ M < N, [N, M] = [DZj(N), M] ≤ [D, M][Zj(N), M] ≤ M, hence M ⊳ N by 5.0.3, and N/M is ′ abelian, being isomorphic to a factor group of Zj(N)/Zj−1(N). It follows that N ≤ M, implying N =(H ∩ N)N ′ = DN ′ ≤ M, a contradiction. To prove (2), suppose ﬁrst that a minimal abelian normal subgroup A has a complement H in G. If H ≤ S

5.11 Theorem. Let G be a group, N G, N ﬁnite, D := N ∩ Φ(G). The following are equivalent: G

(i) N is nilpotent, N

(ii) N/D is abelian, Φ(G) ﬁnite  D (iii) N/D is nilpotent.   In particular, if N ≤ Φ(G), then N is nilpotent. 

Proof. (i) implies (iii) by 5.1.6, and (iii) implies (ii) by 5.10(1) as NΦ(G)/Φ(G) ∼= N/D. G Suppose (ii) and let P ∈ Syl(N), M := DP . As N/D is abelian and N M M/D ∈ Syl(N/D) we have M/D ≤ N/D. It follows that M G. NG(P ) char D Since P ∈ Syl(M), 1.12 implies that G = M NG(P ) = D NG(P ), hence G = Φ(G)N (D). By 5.9.4, N (P ) cannot be contained in P G G any maximal subgroup of G. But |G : NG(P )|≤|D|≤|N| is ﬁnite.  Hence NG(P )= G, P G. Now (i) follows from 5.3. If N ≤ Φ(G), then (ii) and (iii) hold trivially, hence (i).

5.12 Corollary. Let G be a ﬁnite group.

(1) (Frattini 1885) Φ(G) is nilpotent.

(2) (Wielandt 1937) G is nilpotent ⇔ G′ ≤ Φ(G).

(3) (Gasch¨utz 1953) G is nilpotent ⇔ G/Φ(G) is nilpotent.

(4) (Gasch¨utz 1953) F (G)/Φ(G)= F (G/Φ(G)). F (G)/Φ(G) is abelian, is the product of all minimal abelian normal subgroups of G/Φ(G) and has a complement in G/Φ(G).

64 Proof. (1) is clear by the last assertion in 5.11. (2) follows from the equivalence of (i) and (ii), (3) from the equivalence of (i) and (iii) in 5.11, putting N := G. (4) By (1) we have Φ(G) ≤ F (G). Moreover, 5.1.6 implies that F (G)/Φ(G) ≤ F (G/Φ(G)). The converse inclusion holds because (iii) implies (i) in 5.11, where we let N be the normal subgroup of G such that Φ(G) ≤ N, N/Φ(G) = F (G/Φ(G)). By 5.10(1), F (G)/Φ(G) is abelian. For the remaining assertions we may assume w. l. o. g. that Φ(G) = . Let N be a maximal product of minimal abelian normal subgroups of G such that N has a complement K in G. Clearly, N ≤ F (G). Assume F (G) ∩ K = . By 5.9.8, F (G) ∩ K G so that there exists a minimal normal subgroup A of G in F (G) ∩ K. By 5.10(2), A has a complement in G, hence also in K. A complement of A in K, however, is a complement of NA in G, in contradiction to the choice of N. It follows that F (G)∩K = , N = F (G). Thus F (G) is the product of all minimal abelian normal subgroups of G.

5.13 Theorem (Roquette 1964). Let G be a group, π a set of primes, N G, N ﬁnite [π] and N = N. Suppose H ≤ G such that H ∩ N ∈ Hallπ(N). If H ∩ N ≤ Φ(H), then H ∩ N = .

G Proof. Put M := H ∩ N. Then the hypotheses of 4.8 (with M ∗ = M ′)

H are satisﬁed. As M ≤ H ≤ NG(M), there exists a subgroup S of H ′ N Φ(H) such that SM = H, S ∩ M = M . Since |H : S| is bounded by |M|, ′ M hence ﬁnite, 5.9.4 implies S = H. We thus obtain M = H ∩ M = M ≤ Φ(H). By 5.11, M is nilpotent so that 5.1.4 implies M = . 

57 p Special case: Let G be a ﬁnite group, p a prime, P ∈ Sylp(G) and P ∩O (G) ≤ Φ(P ). Then P ∩ Op(G)= .

5.14 Corollary. Let H be a ﬁnite group. Then every prime divisor of |CAut H (H/Φ(H))| 58 is a divisor of |Φ(H)|. If H is a p-group, then CAut H (H/Φ(H)) is a p-group.

Proof. Assume that p ∤ |Φ(H)| for some prime divisor p of |CAut H (H/Φ(H))|. By 1.9(4), CAut H (H/Φ(H)) has a subgroup A of order p. Let G := H ⋊ A, N := [H, A]A. id G Then [H, A] ≤ Φ(H) ≤ H, hence N G,[H, A] = H ∩ N ∈ Hall(N) ′ as p = |N :[H, A]| ∤ |[H, A]|. We have A ≤ N [p ], hence H N Φ(H) [p′] [p′] [H, A] ≤[H, N ] ≤ H ∩ N ≤ H ∩ N ≤ [H, A] A [H,A] p [p′]  so that N = N. Now 5.13 implies that H ∩ N = , i. e., [H, A]= , a contradiction since H cannot be centralized by a nontrivial subgroup of Aut H.

57Clearly, this also follows easily from 4.9(1). 58P. Hall (1933) proved the following stronger result: If H has a system of d generators, then d |CAut H (H/Φ(H))| is a divisor of |Φ(H)| (see [H], III, 3.17 for a proof).

65 5.15 Proposition. Let G be a group, A ≤ Aut G. Suppose that A and [G, A] are ﬁnite and of coprime orders.

(1) G = CG(A)[G, A].

(2) If G is abelian, then G = CG(A)×˙ [G, A]. In particular, (1), (2) hold if G is a p-group and A a p′-subgroup of Aut G. If then [G, A] ⊆ Φ(G), it follows that A = . Proof. (1) follows from 3.14, putting H := [G, A]. α ϕ −1 α βα (2) Let ϕ : G → G, g → α∈A g . Then ϕ ∈ End G and [g, β] = α∈A(g ) g =1G |A| ϕ for all g ∈ G, β ∈ A. For x ∈ CG(A) ∩ [G, A] it follows that x = x =1G which under our hypothesis implies x =1G.

If G is a p-group and A ≤ Aut G such that [G, A] ⊆ Φ(G), then A ≤ CAut G(G/Φ(G)), hence A is a p-group by 5.14. We illustrate 5.15(2) by a typical application (cf. p. 51): 5.16 Corollary. Let G be a finite group, p a prime. Suppose that a Sylow p-subgroup [p] P of G is abelian. Then CP (NG(P )) is a complement of G in G. ′ Proof. By 4.4, P ∩ G = [P, NG(P )] = [P, A] where A is the image of NG(P ) in Aut P with respect to the action of NG(P ) on P by conjugation. We have NG(P ) ≥ CG(P ) ≥ ′ P ∈ Sylp(G), hence A, being isomorphic to NG(P )/CG(P ), is a p -group. By 5.15(2), [p] [p] [p] P = CP (A)×˙ [P, A]. It follows that G = PG = CP (A)G and CP (A) ∩ G ≤ CP (A) ∩ [P, A]= . As CP (A)= CP (NG(P )), the proof is complete. In p-groups, the Frattini subgroup may be characterized in a more specific way than in arbitrary finite groups as will become clear in a moment. For any finite group and any prime p there is a smallest normal subgroup the quotient of which is an elementary abelian p-group: Any such subgroup must contain G′ and the set Gp of all gp (g ∈ G). The latter is generally not a subgroup, but in an abelian group it is. It follows that G′Gp is a subgroup of G, and clearly normal with an elementary abelian factor group. Hence we have 5.16.1. Let G be a finite group, p a prime, N ≤ G. Then N G and G/N is an elementary abelian p-group if and only if G′Gp ≤ N. Furthermore, Φ(G) ≤ G′Gp, the last assertion being a consequence of 5.9.1. Every abelian group M is a Z-module with respect to the natural action M → M ϕ : Z → End M, z → g → gz p If M =  and x = 1M for all m ∈ M, it follows that ker ϕ = pZ, hence ϕ induces an action of the prime field Fp := Z/pZ on M so that M becomes a vector space over Fp. d The Fp-subspaces of M are just the subgroups of M. If M is finite and |M| = p where d ∈ N0, then d = dimFp M.

66 5.17 Burnside’s basis theorem. Let p be a prime and G a p-group.

(1) Φ(G)= G′Gp.

(2) Any two minimal sets of generators of G are equipotent. Their cardinality is the number d such that |G/Φ(G)| = pd.

Proof. (1) Φ(G) ≤ G′Gp by 5.16.1. By 5.2(2), every maximal subgroup of G is normal, hence of index p as the factor group has no nontrivial subgroup. Thus Gp ⊆ Φ(G). As G′ ≤ Φ(G) by 5.12(2), the claim follows. (2) Let X be a minimal set of generators of G. The mapping

X → G/Φ(G), x → Φ(G)x, is injective because y ∈ Φ(G)x, x, y ∈ X, x = y, implies (X {y}) ∪ Φ(G) = G, hence X {y} = G by 5.9.5, a contradiction. Furthermore, the set of generators {Φ(G)x|x ∈ G} of G/Φ(G) is minimal, by 5.9.6. Thus any minimal set of generators of G maps bijectively onto a basis of the Fp-vector space G/Φ(G) which is of dimension d. The claim follows.

67 A Appendix and Outlook

Recall that the class of all soluble groups is closed with respect to extensions: If N is a soluble normal subgroup of a group H such that H/N is soluble, then H is soluble. In particular, a product of ﬁnitely many soluble normal subgroups is a soluble normal subgroup59. As in the case of the class of nilpotent groups (cf. 5.7), this observation gives rise to the following deﬁnition:

A.1 Deﬁnition. Let G be a group. Then

Sol(G) := N NEG Nsoluble is called the soluble radical of G. If G is ﬁnite, then Sol(G) is soluble and Sol(G/Sol(G)) =  because the class of all soluble groups is closed with respect to extensions. Thus there is a natural general programme to study ﬁnite groups, consisting of the following three parts:

• Describe all soluble groups

• Describe all groups with trivial soluble radical

• Describe all extensions of soluble groups by groups with trivial soluble radical.

We will add a number of observations about each of these three points. As a result we will see which topics constitute key chapters in the pursuit of the above programme. This may serve as a help for an orientation about current research efforts in finite group theory. The socle of G is defined by Soc(G) := M. M G min If G is finite and soluble, every chief factor of G, being a minimal normal subgroup of a quotient of G, is elementary abelian. Furthermore, 5.12(4) implies the following remark:

A.1.1. If G is a ﬁnite soluble group, then Soc(G/Φ(G)) = F (G)/Φ(G).

59In other words, the class of soluble groups is closed with respect to ﬁnite normal products. Note that this remark is almost trivial in contrast with the analogous assertion for the class of nilpotent groups (see 5.6).

68 From 2.12 we know that in the ﬁnite case a minimal normal subgroup M has the form S ×× S for some simple group S and some n ∈ N. n

n ∼ (Aut S) Aut M If S is nonabelian, then Aut M = Sn ≀ Aut S, by 3.2(2). The normal id (In S)n subgroup In S of Aut S is isomorphic to S. Hence Aut M is an ex- ∼ tension of a normal subgroup = M and Sn ≀ Out S where Out S := Sn id  Aut S/In S, called the outer automorphism group of S.60 For example, let S = An for some n ≥ 5. By 1.3, S is simple. As An Sn, the action of Sn on An given by conjugation is a monomorphism of Sn into Aut An. It can be shown that this monomorphism is an isomorphism unless n = 6. Thus, for n = 6, |Out An| = 2. Furthermore, |Out A6| = 4 (see, for example, [H], II, 5.5).

A.2 Proposition. Let G be a ﬁnite group such that Sol(G)= . Let M1,...,Mk be the (mutually distinct) minimal normal subgroups of G. (Then Soc(G) = M1×˙ ×˙ Mk as the Mi are non-abelian.) Let

f : G → (Aut M1) ×× (Aut Mk), g → (¯g|M1 ,..., g¯|Mk ), where g¯ denotes the conjugation by g. Then f is a monomorphism.

Proof. Each Mi is invariant under conjugation in G, hence f is a homomorphism and ker f = CG(Soc(G)). Assume that ker f = . Then ker f contains a minimal normal subgroup of G, w. l. o. g. M1 ≤ ker f. Thus [M1, M1] ≤ [Soc(G),CG(Soc(G))] = , i. e., M1 is abelian, a contradiction. Hence ker f = , f is injective. In the sense of A.2, the structure of G is controlled by the socle of G if G is finite and Sol(G)= : The groups G with a given socle (more precisely, with prescribed minimal normal subgroups M) may be found as subgroups of the direct product of the groups Aut M the structure of which was considered above, if G is finite and Sol(G) = . Of interest are the subgroups which contain In M for every M and act transitively on the set of simple direct factors S of M. The main unknown in this approach is the structure of Aut S. Roughly spoken, the description of all finite groups G with Sol(G) =  along the above lines depends on the knowledge of all finite simple groups S and their automorphism groups. We have already seen (cf. 5.8.1 and the subsequent comments) that the structure of a finite soluble group G is controlled in a similar way by F (G): Given any finite nilpotent group N, the finite soluble groups G such that F (G) = N have the property that G/Z(N) may be found, up to isomorphism, as subgroups of the direct product of the groups Aut P containing In P as their largest normal p-subgroup, where P ranges over

60The famous Schreier conjecture claims that the outer automorphism group of a finite simple non-cyclic group is always soluble. Having been undecided for decades, this has been proved – like a number of other long-standing conjectures in group theory – on the basis of the classification of all finite simple groups.

69 Syl (N). In this sense, the description of all finite soluble groups allows a reduction to the study of all p-groups and their automorphism groups, plus the extension problem of an abelian normal subgroup V (∼= Z(N)) by a finite soluble group U (∼= G/Z(N)). The general extension problem asks for a description of all groups which have a given normal subgroup V with a likewise given quotient U. The best understood extensions are the splitting ones, given by the semidirect products of V and U, introduced and studied in Chap. 3. Therefore, the split extensions of V by U may be viewed as the trivial cases of the general extension problem. They depend on the possible group actions of U on V and nothing more. Conditions on the groups U, V which guarantee that every extension of V by U must necessarily split constitute therefore a most satisfactory and useful contribution to the extension problem, solving it completely in those cases. The famous theorems 3.9 and 3.13 are results of this kind. But without restrictive hypotheses on V and U there will, in general, exist non-split extensions which are not accessible by the methods treated in Chap. 3. An interesting connection between the field of all extensions (of two given groups) and that of split extensions is, however, established by means of the wreath product as has been proved in 3.7. A major special case, for many problems the real core, is that of an abelian normal subgroup. In particular, we have seen this to be true with respect to the aspect of a finite soluble group as an extension of the centre of its Fitting subgroup. Moreover, the problem of extending a soluble normal subgroup V may be viewed as an iteration of extending abelian normal subgroups (in the first step extending V/V ′ by the desired quotient, then V ′/V ′′, by the quotient obtained in the foregoing extension step, etc.). Thus we also have a reduction of the third point in the list on p. 68 to the case of an abelian normal subgroup. Let G be an extension of an abelian group V by a group U. Then V , being abelian, is contained in the kernel of the action of G on V . Hence each such extension G is associated with some action of U on V . As in the case of the semidirect product we may thus start from a given action of U on V and restrict ourselves to the study of possible extensions G which induce that particular group action of U on V . Thus we view V as an U-module, fixing a certain action of U on the abelian group V right from ∼ ∼ the beginning. A tiny example is given by V = U = Cp for a prime p, where the only possible action is the trivial one. Clearly, each extension of V by U is of order p2 and abelian (cf. also 1.8.1). In this extremely simple special case of 2.11 there are exactly two distinct isomorphism types: the cyclic one and the elementary abelian one. But even in this little example we see already that there is not only the split extension (the elementary abelian group) but also a non-split one (the cyclic group). Recall that this cannot happen if gcd(|U|, |V |) = 1, by 3.13. But as soon as this hypothesis of coprime orders is not satisfied we have to expect a nontrivial situation, i. e., more than just the splitting extension of V by U. A simple example with nontrivial group action is given by ∼ ∼ −1 V = C4, U = C2, where the nontrivial element of U induces the automorphism v → v of V . The extensions with respect to this group action are represented by the dihedral group D8 (the split extension) and by the quaternion group Q8 (a non-split extension).

70 An important tool for investigations about modules of a group is cohomology theory. The so-called n-th cohomology group with respect to an action f of a group U on an abelian group V is a certain factor group of a subgroup of (V U n , +)˙ (see 2.3). For example, if n = 1, this subgroup is given by the crossed homomorphisms of U into V (see 3.3). One −1 uf readily verifies that for every v ∈ V the mapping wv : U → V, u → v v , is a crossed homomorphism. The factor group of the group of all crossed homomorphisms by the subgroup consisting of the mappings wv (v ∈ V ) is the first cohomology group. The second cohomology group (n = 2) arises in a more complicated but principally similar way. The important link with the extension problem is the fact that each of its elements allows an interpretation as a group extension of V by U (respecting the given action f) and that all of these extensions are obtained by means of (at least) one element of the second cohomology group. Its neutral element corresponds to the semidirect product of V by U with respect to f. Hence the above remark that the latter may be viewed as the trivial case of an extension of V by U has a concrete structural background. To study extensions therefore means to determine the second cohomology group. This may look like an isolated detail at the first glance, but in fact there are strong connections between the cohomology groups for different n ∈ N so that the whole theory plays a role even if in some context (like here) only a particular cohomology group is of interest. Summarizing, we may consider the following topics as important chapters in finite group theory in regard of the programme sketched on p. 68:

• p-groups and their automorphisms groups,

• ﬁnite simple groups and their automorphism groups,

• cohomology theory (in particular: extension theory).

The remarks in this appendix result in a modified version of what may be viewed as Hans Fitting’s legacy, his programmative paper [F]. Since the days of Fitting, enormous progress has been made with respect to the second point. If we may trust in what has been claimed by experts of the area of simple groups for many years, the classification is complete. While [W] is a detailed description of all finite simple groups, hence is a resultative account after the efforts of several decades of hardest work, the project of a complete presentation of a proof of the classification itself, however, is still unfinished although apparently well on its way. On the other hand, the first and the third of the above-mentioned points are still far from a satisfactory understanding although much energy has already been put into those areas too. Thus group theory is and will probably remain for a long time a research area with deep secrets to be discovered. Moreover, it should be clear that Fitting’s programme and also a modernized version of it is only one possible kind of orientation in this fascinating universe. From a certain point of view it may help to understand main streams in group theory. But group theory is certainly a more complex field. The reader should bear in mind that the importance of groups lies, to a considerable extent, in their occurrence in other mathematical theories. While it is true that group theory has become a highly elaborate and specialized building of

71 notions and arguments, it is likewise true that those connections have had an important inﬂuence on group theory. In this course we have presented purely group-theoretic methods and concepts which allow to penetrate into many directions of modern research. Our presentation should not be interpreted as a narrow-minded limitation to a so-called purely group-theoretic position. But for a fruitful combination with other areas inside or outside Algebra, a thorough knowledge of the rich spectrum of purely group-theoretic concepts is, in our opinion, the best preparation.

72 Bibliography

[B] Baer, Reinhold, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 (1957), 256–270.

[FT] Feit, Walter and Thompson, John G., Solvability of groups of odd order, Paciﬁc J. Math. 13 (1963), 775–1029.

[F] Fitting, Hans, Beitr¨age zur Theorie der endlichen Gruppen, Jber. Dtsch. Math.- Ver. 48 (1938), 77–141.

[G] Gorenstein, Daniel, Finite Groups, Harper & Row, 1968

[GI] Gagola, Jr., Stephen M. and Isaacs, I. M., Transfer and Tate’s theorem. Arch. Math. 91 (2008), 300-306.

[H] Huppert, Bertram, Endliche Gruppen I, Springer, 1967

[I] Isaacs, I. Martin, Finite Group Theory, Grad. Stud. in Math. 92, AMS, 2008

[KS] Kurzweil, Hans and Stellmacher, Bernd, The Theory of Groups, Springer, 2004

[L] Lam, T. Y., A First Course in Noncommutative Rings, Springer, 20012

[R] Robinson, Derek J. S., A Course in the Theory of Groups, Springer, 19962

[W] Wilson, Robert A., The Finite Simple Groups, Springer, 2009

73 Index

Ω-decomposition, 26 crossed homomorphism, 39 cycle, 6 action, 12 cycle decomposition, 7 ascending chain condition, 29 automorphism, 5 derived series, 45 automorphism descending chain condition, 29 inner, 5 dihedral group, 35 automorphism group, 5 direct Ω-decomposition, 24 automorphism group direct Ω-factor, 24 inner, 6 directly Ω-indecomposable, 24 disjoint cycles, 7 bipartite Ω-decomposition, 24 elementary abelian, 31 central chain, 55 Engel element, 60 central series equivalent sets under an action, 15 ascending, 55 extra-special, 62 descending, 55 centralized group action, 17 factor, Ω-, 24 centralizer, 14 Fitting subgroup, 59 centre, 5 focal subgroup, 50 chain Frattini subgroup, 62 ascending, 55 fully invariant, 23 descending, 55 characteristic, 23 group chief factor, 32 π, 18 chief series, 32 -p, 18 commutator subgroup, 23 -p′, 18 compatible, 15 alternating, 8 complement, Ω-, 24 simple, 9 complete wreath product, 37 group action, 12 composition factor, Ω-, 32 group action composition series, Ω-, 32 elementwise ﬁxed-point-free, 13 conjugacy class, 6 faithful, 12 conjugation, 5 transitive, 13 constrained, 61 group with operators, 12 control of transfer, 52 Hall subgroup, 45 core, 14

74 holomorph, 36 semidirect decomposition, 25 homomorphism, Ω-, 24 semidirect product, 34 hypercentre, 55 sign homomorphism, 8 similar sets under an action, 15 inverse left multiplication, 13 similar subnormal series, 32 left Engel element, 60 simple, Ω-, 24 left-normed commutator, 58 socle, 68 length of a cycle, 7 space length of a subgroup series, 31 K-, 23 split extension, 35 module, 23 stabilizer, 9 monomial matrix, 38 step number, 45 subgroup nilpotency class, 56 maximal, 9 nilpotent group, 55 subnormal, 31 nilpotent residual, 57 subnormal series, 31 non-generators, 63 Sylow subgroup, 20 normal series, 31 normal set of operators, 26 transfer, 48 normal subset, 6 transposition, 8 normalized group action, 16 transvection, 11 normalizer, 14 type of an abelian p-group, 30 operating system, 12 uniquely n-divisible, 44 operator, 12 orbit, 13 weakly noetherian, 62 orbit wreath product, 36 π, 6 nontrivial, 6 outer outomorphism group, 69 partition of a positive integer, 30 permutation even, 8 permutation matrix, 38 regular wreath product, 37 replacement automorphism, 26 representation of a K-space, 23 of an algebra, 24 retracting Ω-factor, 27 right Engel element, 60 right multiplication, 13 Schreier conjecture, 69