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Group Theory Group Theory 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 (Gasch¨utz), 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.
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