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1. actions and other topics in

October 11, 2014

The main topics considered here are group actions, the , semi-direct products, nilpotent and solvable groups, and simple groups. See Preliminary remarks for some of the notation used here, especially regarding general linear groups. Some further notation: [n] denotes the of the first n natural numbers 1, 2, ..., n. Pk[n] denotes the set of k-element of [n].

1 Group actions

1.1 Definition of a or G-set Let G be a group, with identity element e.A left G-set is a set X equipped with a map θ : G × X−→X satisfying (i) θ(gh, x) = θ(g, θ(h, x)) for all g, h ∈ G and all x ∈ X, and (ii) θ(e, x) = x for all x ∈ X. Usually we write either g · x or simply juxtaposition gx for θ(g, x); in the latter notation conditions (i) and (ii) become (gh)x = g(hx) and ex = x. We also call this data a group action, or say that “G acts on X” (on the left). Similarly a right G-set is a set X equipped with a map θ : X × G−→X satisfying (in the evident juxtaposition notation) x(gh) = (xg)h and xe = x. Of course the distinction between G-actions does not depend on whether we write the domain of θ as G × X or X × G. The distinction is that in a left action gh acts by h first, then g, whereas in a right action g acts first, then h. Notice that this point, we haven’t even used the existence of inverses, so exactly the same definition makes sense for left and right actions. We will make little use of monoid actions, however. One immediate advantage of the existence of inverses is that any right action can be converted to a left action by setting g · x = xg−1; similarly any left action can be converted to a right action. Nevertheless it is important to pay close attention to which side the group is acting on. If the side is not specified we always mean a left action (an arbitrary choice on my part!). Any statement about left actions has a parallel statement for right actions; we leave it to the reader to make the translation.

1.2 Some fundamental terminology Let X,Y be (left) G-sets. A map φ : X−→Y is a G-map or a G- if φ(gx) = gφ(x) for all g ∈ G, x ∈ X. Then G-sets and G-maps form a that we will denote

1 G-Set.

The fixed-point set XG is defined by XG = {x ∈ X : gx = x ∀g ∈ G}. This defines a G-set −→ Set in the evident way. The group Gx of a point x ∈ X is defined by Gx = {g ∈ G : gx = x}; clearly Gx is a . Thus the fixed-points are the points x with Gx = G. The action is trivial if every point is a fixed point. At the opposite extreme, the action is free if Gx = {e} for all x ∈ X.

Define an on X by x ∼ y if there exists g ∈ G such that gx = y.A is called an , usually denoted O. The orbit determined by a particular x ∈ X is denoted Ox or Gx. The set of all orbits of a left action is denoted G\X; the set of orbits of a right action is denoted X/G. This notational distinction is important because we will often have groups acting on the left and the right of the same set X. The action is transitive if there is only one orbit. In other words, for all x, y ∈ X there exists a g such that gx = y. Transitive actions will be discussed in more detail in a later section. If O ⊂ X is an orbit and x ∈ O, then the map G−→X given by g 7→ gx factors through =∼ a G-bijection G/Gx −→ O = Gx. Hence [G : Gx] = |Gx| (this is true even when the sets in question are infinite, but we have in mind here the finite case).

Finally, we recall a simple but very powerful counting formula. Suppose the group G acts on the finite set X. Then

X X |X| = |O| = [G : Gx], O x∈G\X where the first sum is over the orbits O of the action and the second sum means, in a mildly abusive notation, that we are taking a fixed representative x of each orbit. This choice of x ∈ O is arbitrary, but the sum is nevertheless well-defined since [G : Gx] = |O| is independent of the choice.

2 Examples

1. The Sn acts on the left of [n] := {1, 2, ..., n} by permutions. The action is transitive, with the isotropy group of any point isomorphic to Sn−1. More generally, if X is any set, we let P erm X denote the group of bijections X−→X. Then by construction P erm X acts on the left of X by σ · x = σ(x). It is a left action because a composition σ ◦ τ acts by τ first, then σ.

2. If G is any group, H any subgroup, then the left translation action of H on G is defined by h · g = hg for h ∈ H, g ∈ G. The right translation action is given by g · h = gh. These are both free actions. The orbit H\G of the left action is by definition the set of right Hg, while the orbit space G/H of the right action consists of the left cosets gH. If G is finite, then (since every orbit has size |H|) the counting formula just says that |G| = |H| · [G : H].

2 3. If G is any group, the left conjugation action of G on itself is given by g · x = gxg−1. Similarly right conjugation is defined by x · g = g−1xg. The fixed-point set of either action is the C(G)(Z(G) is another common notation for the center). The orbits are the conjugacy classes of G. The isotropy group of x is CGx, the centralizer of x in G. In this case the counting formula yields the class equation. To state it we need a notation for conjugacy classes, and—sadly—we have already assigned the letter “C” to centers and centralizers. I will use the non-standard notation κ(x) to mean the of x, and Conj G to mean the set of conjugacy classes. We then have

X X |G| = |κ(x)| = [G : CGx]. x∈Conj G x∈Conj G Once again the notation has the obvious interpretation: We are choosing one x from each conjugacy class, and the choice doesn’t matter.

4. If G is any group, let S(G) denote the set of of G. Then G acts on S(G) by left conjugation: g · H = gHg−1 (there is also a right conjugation, of course). The fixed-points are the normal subgroups. The orbits are conjugacy classes of subgroups. The isotropy group of H is the normalizer NGH of H in G.

5. Let X,Y be sets and let F (X,Y ) denote the set of functions X−→Y . If Y is a left G-set, we get a left G-action on F (X,Y ) by (g · φ)(x) = g(φ(x)). If X is a left G-set, we get a right G-action on F (X,Y ) by (φ · g)(x) = φ(gx). Note carefully that this is a right action. However, we can always convert it to a left action by (g ? φ)(x) = φ(g−1x). If both X and Y are G-sets, we can get a combined left action of G on F (X,Y ) by (g · φ)(x) = gφ(g−1x). The fixed-point set of the “combined” left action gφg−1 is the of G-equivariant maps X−→Y , as is easily checked.

n 6. Let X be a set, X the n-fold of X with itself. Then Sn acts on Xn by permuting the coordinates. This is a right action, given explicitly by

(x1, ..., xn) · σ = (xσ(1), ...xσ(n)). One could check directly that this is a right action, but easier is to note that Xn = F ([n],X), where Sn acts on the domain, on the left. So this is a special case of the previous example. The fixed-point set is the diagonal subset of all (x, x, ..., x). Contemplation of the orbits and isotropy groups is left to the reader.

7. Projective spaces. The purpose of this example is twofold. First of all, projective spaces are ubiquitous in and —including especially algebraic geometry, which leads me to discuss them in an algebra course. Second, in “nature” we are often first confronted not with a group action or even a group, but with a set (or , etc.) X that may secretly be equipped with a useful group action and/or realization as the orbit set of a group action. It’s important to be able to recognize such structures. Let F be a field, and let V be a finite dimensional over F . (In fact the finite-dimensionality isn’t necessary, but I prefer to avoid distractions.) The P(V ) is the set of lines through the origin in V . Even though at the moment we are not giving

3 it any topology, we call it a “space” because that is the traditional term, and if you call it the “projective set” you risk being sneered at as an ignorant yokel from the backcountry. Thus P(V ) does not involve a group in its definition, but it is in fact crawling with groups. First of all, it is the orbit set of the action of F × on V − {0} by . This is a very handy interpretation. Second, GL(V ) acts transitively on it: By elementary linear algebra, any can be moved to any other line by an invertible linear transformation. This is a particularly important interpretation; it is often useful to recognize a set as a transitive G-set (also known as a “”). To complete the picture we choose a convenient point in the set and determine its isotropy group. Here there is no natural choice of a line; we just pick one and call it L0. The isotropy group H consists of invertible transformations preserving L0, i.e. the set of A ∈ GL(V ) such that A has an eigenvector in =∼ L0. Then there is a G- GL(V )/H −→ P(V ). If we want to be more explicit, we choose a basis e1, ..., en and take L0 = he1i. Then H corresponds to the group of matrices with ai1 = 0 for i > 1 (i.e. the first column is zero except for a11).

3 Actions preserving some additional structure

Frequently, the G-sets X one encounters are not merely sets but have some extra structure preserved by the action. Some examples:

Example. X is itself a group, and G is acting on it via group automorphisms. In other words, g · (xy) = (g · x)(g · y) for all g ∈ G, x, y ∈ X. The action of G on itself by conjugation is an action of this type. Actions via group automorphisms will be used to construct “semi-direct products” later.

Example. We have a vector space V over a field F , and the G-action is linear: g · (v + w) = g · v + g · w, and g · (cv) = cg · v (c ∈ F ). This type of action is called a representation of G over F. In this course “representation” will always be taken to mean finite dimensional representation, unless otherwise specified. Note that GL(V ) acts linearly on V by definition; we call this the standard representation of GL(V ). is one of the major branches of mathematics. We’ll consider representation theory of finite groups in some detail, especially over C.

Example. Let F be a field, and suppose G acts on F via field automorphisms. This is precisely the situation one studies in Galois theory.

Example. Let X be a topological space, and suppose G acts on X via homeomorphisms. In other words, for each fixed g, the map x 7→ g · x is a homeomorphism. In fact we only need to check this map is continuous, since it automatically has an inverse given by the action of g−1. In the topological case, however, G itself might be a —i.e. both a space and a group, with the multiplication and inverse maps continuous. In that context a “topological group action” means a group action such that the action map G × X−→X is continuous (G × X has the ). This is usually a much stronger assumption than merely saying that G is acting by homeomorphisms.

4 There are many variants on this theme: groups acting on metric spaces by , smooth Lie groups actions on smooth (studied in our “Manifolds” course), sim- plicial actions on simplicial complexes...all of these are dear to my heart, but lie outside the scope an algebra course.

4 An alternate view of group actions

Let X be a G-set. Then each g ∈ G defines a bijection X−→X, so we get a map ρ : G−→P erm X. Moreover, the axioms for a group action translate into the statement that ρ is a . Conversely if a homomorphism ρ : G−→P erm X is given, we get a G-action on X by g · x = ρ(g)(x). Thus G-actions on X are the same thing as homomorphisms G−→P erm X. If ρ is injective we say that the action is faithul (“effective” is another commonly used term). Note that free actions are faithful, but not conversely: the action of Sn on [n] is faithful, but certainly not free. Indeed an action is free if and only if all isotropy groups are trivial, whereas an action is faithful if and only if the intersection ∩x∈X Gx of all the isotropy groups is trivial. In analyzing the structure of a newly encountered group G, optimists hope to find a proper non-trivial H such that H and G/H are already “known”, or at least more approachable. Since kernels are always normal, Ker ρ is at least a candidate. Moreover, even when Ker ρ is trivial (i.e. the action is faithul), thereby sinking the optimists’ strategy, we get an injective homomorphism G−→P erm X that may yet prove useful for understanding G. Let’s consider the case when G is finite and X is a finite G-set. If |X| = n, a choice of ∼ ordering of X yields an isomorphism P erm X = Sn, so we’ll think of ρ as a homomorphism G−→Sn. But to get any use out of this method we first need to find some finite G-sets. The most obvious candidate is G itself, with the left translation action. Since the action is free and hence faithful, this yields an injective homomorphism ρ : G−→Sn. Thus every finite group is isomorphic to a subgroup of some Sn, a result known as “Cayley’s theorem”. The n obtained, however, is n = |G|, and |Sn| is too large for this result to be of more than occasional use. To find smaller G-sets, we can choose a subgroup H and consider X = G/H with its trans- lation action. But how do we find subgroups of a general finite G? The Sylow p-subgroups of G and their normalizers, discussed in the next section, form the most important and most useful family of subgroups that are defined for an arbitrary finite group. Alternatively, one may be able to exploit special features of a particular G. As a simple, fun example, consider 2 GL2F2. It acts linearly on F2, a vector space having a grand total of four elements. So there are three nonzero vectors, and after choosing an ordering of them, we obtain a homomor- phism ρ : GL2F2−→S3. This ρ is especially useful, as it turns out to be an isomorphism (exercise).

Remarks. 1. If X is a G-set with extra structure, then ρ : G−→P erm X lands in the of this structure. In example 1 we get ρ : G−→AutgrpX, in example 2 we get ρ : G−→GL(V ), and so on. Indeed if C is any category, we can define a G-object in the category to be an object X together with a homomorphism G−→AutCX.

5 2. Suppose X1,X2 are G-sets with the same underlying set X but with different G-actions. Let ρ1, ρ2 be the corresponding homomorphisms G−→P erm X. Then X1 is isomorphic to X2 as a G-set if and only if ρ1, ρ2 are conjugate as homomorphisms to P erm X. (Exercise. −1 By “conjugate” I mean there is a σ ∈ P erm X such that σρ1σ = ρ2.) 3. If X is a non-faithful G-set, with H = Ker ρ, then the G-action factors through a G/H-action: (gH)·x = gx. This is well-defined since H is normal. For example, consider the action of GL(V ) on P(V ) discussed earlier. The of this action is the scalar matrices × × × F ⊂ GL(V ), so we get an action of GL(V )/F on P(V ). This is the reason GL(V )/F is called the projective , denoted P GLnF .

5 Transitive G-sets

Recall that a G-set X is transitive if there is only one orbit, and that in this case a choice =∼ of x ∈ X yields an isomorphism of G-sets G/Gx −→ X. Notice, however, that the choice of x is arbitrary. This is true even for a free transitive action: then we get an isomorphism of ∼ G-sets G −→= X, where G has the left translation action, but there is no natural choice of such an isomorphism; it depends on the choice of x. The next result is basic. Proposition 5.1 Let X be a transitive G-set. Then the isotropy groups form a complete conjugacy class of subgroups of G.

−1 Proof: Suppose x, y ∈ X. Choose g ∈ G with gx = y. Then gGxg = Gy, as is readily checked. Hence any two isotropy groups are conjugate. Conversely, let H be a subgroup −1 conjugate to Gx; say gGxg = H. Then H = Ggx, so every subgroup in the conjugacy class occurs as an isotropy group.

Now suppose we have two subgroups K,H and we want to show that K is conjugate to a subgroup of H, a problem that arises quite frequently. In the spirit of the previous proposition we have at once: Proposition 5.2 K is conjugate to a subgroup of H if and only if the left action of K on G/H has a fixed point. More precisely, KxH = xH if and only if x−1Kx ⊂ H. Now let’s consider the set of all orbits of the K-action on G/H. These are called the (K,H)-double cosets, denoted K\G/H. We could just as well think of K\G/H as the H- orbits of the right action on K\G, or more symmetrically as the orbits of the left K×H-action on G given by (k, h) · g = kgh−1. More often, however, we stick with the first interpretation. Here is an example known as the Bruhat decomposition:

Proposition 5.3 Let F be a field, and let B := BnF . Then the (B,B)-double cosets of GLnF are given by

a GLnF = BwB, w∈W ∼ where W = Sn is the Weyl group.

6 The proof is by old-fashioned row and column reduction, and is left to the reader. What’s not obvious is why this particular way of arranging the row/column reduction is of interest. It turns out that it plays an important role in the structure theory of a large class of interesting groups (not just the general linear groups), and enters into algebraic geometry and topology via “Schubert varieties” and “Schubert cells”. For example, when F = R or F = C then the orbit set GLnF/B is a topological space known as a “flag ” or “flag variety”, and the left B-orbits—which by the proposition are indexed by elements of W —are homeomorphic to vector spaces over F and called Schubert cells. They are very useful for studying the geometry and topology of flag manifolds, a subject which is in itself a major industry these days. I mention all this just to pique your curiosity, and to suggest how the humble process of row reduction connects with beautiful, deep mathematics.

6 A fixed-point theorem, and the Sylow theorems

Throughout this section, p is a prime. Everything will be based on the following simple fixed-point theorem:

Theorem 6.1 Let P be a finite p-group, S a finite P -set. Then |S| = |SP | mod p.

Proof: |S| = P |O|, where O ranges over the P -orbits. Since P is a p-group, |O| is either divisible by p or consists of a single fixed point, whence the result.

Corollary 6.2 If |S| is prime to p, then there is at least one fixed point.

As an immediate application of the corollary, we have:

Proposition 6.3 Every finite p-group G has non-trivial center.

Proof: Consider the action of G by conjugation on G − {e}. By the corollary it has a fixed-point, so G has non-trivial center.

The proposition in turn has a nice corollary:

Corollary 6.4 If G has p2 (p a prime) then G is abelian.

Proof: By the proposition, G has a central subgroup H of order p. Hence |G/H| = p, so G/H is cyclic. But whenever a group G has a central subgroup with cyclic , G is abelian (why?).

The next application can be proved using the binomial theorem, but we can also deduce it from Theorem 6.1.

k  n  Proposition 6.5 Let n = sp . Then pk = s mod p.

7 k Proof: Partition [n] into s disjoint subsets A1, ..., As of size p . Let Ci ⊂ Sn be a cyclic k group of order p that permutes Ai transitively and fixes the other Aj’s pointwise, and take P = C1 × ... × Cs. Then the action of P on Ppk [n] has exactly s fixed points, namely A1, ..., As. Hence the proposition follows from the theorem.

Let G be a finite group, and write |G| = spk with s prime to p.A p-Sylow subgroup is a subgroup P of order pk. The first Sylow theorem asserts that such subgroups always exist.

Theorem 6.6 G has a p-Sylow subgroup.

Proof: The strategy is to look for a finite G-set S such that some isotropy group Gx is a p-Sylow subgroup. Indeed, suppose we had a finite G-set S such that (i) |S| is prime to p; and (ii) all isotropy groups Gx are p-groups. Then by (i) there is an orbit O with |O| prime i to p. Choose x ∈ O; then |Gx| = p for some i by (ii). But |G| = |Gx| · |O|, forcing i = k. Hence Gx is a p-Sylow subgroup. It remains to exhibit such an S. Take S to be the set of subsets of G of size pk, with action induced by the left translation action of G on itself. Then (i) holds by Proposition 6.5. Now let A ∈ S. Then GA acts freely on the elements of A (since the left translation action k is free), so |GA| divides p , proving (ii).

The second Sylow theorem says that all p-Sylow subgroups are conjugate. More precisely:

Theorem 6.7 Let H be a p-subgroup of G, P a p-Sylow subgroup. Then H is conjugate to a subgroup of P . In particular, any two p-Sylow subgroups are conjugate.

Proof: As discussed earlier, this is equivalent to saying that the action of H on G/P has a fixed point: if HxP = xP , then x−1Hx ⊂ P and conversely. But H is a p-group and |G/P | = s is prime to p, so this follows immediately from Theorem 6.1.

The last item of business is to say something about the set of all p-Sylow subgroups of G. How many such subgroups are there? By the second Sylow theorem, G acts transitively on this set by conjugation. If we fix a p-Sylow subgroup P , the isotropy group of the action is the normalizer NGP . Hence the number of p-Sylow subgroups is [G : NGP ]. This brings us to the third Sylow theorem:

Theorem 6.8 Let npG denote the number of distinct p-Sylow subgroups of G. Then npG divides |G| and npG = 1 mod p.

Proof: Since npG = [G : NGP ] for any choice of p-Sylow subgroup P , npG divides |G|. Now fix a p-Sylow subgroup P , and note that by the second Sylow theorem P is the unique p-Sylow subgroup of NGP (since it is a normal p-Sylow subgroup of NGP ). Now consider the −1 left translation action of P on G/NGP . If P xNGP = xNGP then x P x ⊂ NGP , forcing −1 x P x = P since P is the unique p-Sylow subgroup of NGP . Hence x ∈ NGP ; in other words, the P -action has a unique fixed point. Using Theorem 6.1 we conclude

8 P npG = |G/NGP | = (G/NGP ) = 1 mod p.

Here’s another interesting fact about p-Sylow subgroups:

Proposition 6.9 Let P be a p-Sylow subgroup of G, and suppose NGP ⊂ H. Then NGH = H. (In particular, this is true for H = NGP .) Proof: Suppose gHg−1 = H. Then gP g−1 is a p-Sylow subgroup of H, so by the second −1 −1 −1 Sylow theorem there is an h ∈ H such that hP h = gP g . Then h g ∈ NGP , so g ∈ H. Many examples and applications of p-Sylow subgroups can be found in the exercises.

7 New G-sets from old: restriction, disjoint unions, products, and induction

Change of group and restriction. Suppose X is a left G-set and φ : H−→G a homomorphism. Then X is a left H-set with action h · x = φ(h)x. Thus φ defines a functor φ∗: G-set −→ H-set. The case when φ is inclusion of a subgroup is of particular importance; in this case we use the notation XH for X regarded as an H-set, and call X 7→ XH the restriction functor. Disjoint unions. Suppose X and Y are G-sets. The X ` Y is a G-set in the obvious way, and in fact is the categorical coproduct. Similarly if Xα is any collection of G- ` sets indexed by a set J, α∈J Xα is a G-set and is the categorical coproduct. The fixed-point set and orbit set take coproducts of G-sets to coproducts of sets.

Products of G-sets. Again let X,Y be G-sets. The product X ×Y is a G-set via the diagonal action: g · (x, y) = (gx, gy). The product of any collection of G-sets is defined similarly, and is the categorical product. The fixed-point functor takes products to products, e.g. (X × Y )G = XG × Y G. Orbits, however, are another matter; there is no simple relationship between G\(X × Y ) and G\X, G\Y . We will see some examples later.

Balanced products. Suppose X is a right G-set, Y a left G-set. Define an equivalence relation on X ×Y by (xg, y) ∼ (x, gy). The balanced product X ×G Y is the set of equivalence classes. In fact this is just the orbit set of the left G-set X ×Y , where g acts by g ·(x, y) = (xg−1, gy). But the slight change in viewpoint can be useful and enlightening. For our immediate purposes, the “induced” G-spaces below provide the most important example.

Induction. Suppose H is a subgroup of G and X is a left H-set. Then the balanced product G×H X is a left G-set with action g1 ·[g, x] = [g1g, x]; we say that the G-action is induced from the H-action. Here the brackets [] denote equivalence class in the balanced product. Thus X 7→ G ×H X defines the induction functor H-set −→ G-set (the definition of the functor on being obvious). Note that the map i : X−→G ×H X given by i(x) = [e, x] is an H-map. Induction has the following universal property (see the notes for a general discussion of such properties). We keep the above notation.

9 Proposition 7.1 Suppose Y is a G-space, X is an H-space and φ : X−→Y is an H-map. Then there is a unique G-map ψ : G ×H X−→Y such that the following diagram commutes:

i - X G ×H X

p φ p p ∃!ψp p p ?© p p p Y p p

Proof: There is no choice in the definition of ψ: We must take ψ([g, x]) = gφ(x). Now check that it works (in particular, check that ψ is well-defined).

As usual, we give two alternate ways of thinking about the universal property:

Plain English version. If you want to define a G-map G ×H X−→Y , it is enough (indeed equivalent) to define an H-map X−→Y .

Adjoint functor version. Induction H-set −→ G-set is left adjoint to restriction G-set −→ H-set. See the category theory notes for discussion of adjoint functors. In essence there is not much to it in the present example; the universal property translates immediately to the assertion that there is a bijection

∼ HomG(G ×H X,Y ) = HomH (X,Y ). To complete the proof that these are adjoint functors, one has to show that the above bijection is natural in X and Y . This is easy once one has absorbed the definition of “”, but it is not essential to understand all this right away. Just use the universal property.

There is a recognition principle for induced G-sets. Let Y be a G-set, X ⊂ Y an H- subset (H ⊂ G). Applying the universal property to the inclusion j : X−→Y , we get a G-map ψ : G ×H X−→Y ; indeed it is just ψ([g, x]) = gx.

Proposition 7.2 ψ is bijective if and only if (i) for all y ∈ Y , there is a g ∈ G such that gy ∈ X; and (ii) whenever x1, x2 ∈ X and gx1 = x2, we have g ∈ H.

The proof is a straightforward check; (i) gives the surjectivity and (ii) the injectivity. Note that (ii) says that if g∈ / H, then g moves every element of X to an element not in X.

n Example. Let F be a field, G = GLnF , and Y the set of pairs (L, v) with L a line in F and v ∈ L. We have an evident G-action on Y given by g · (L, v) = (gL, gv). Let L0 denote the line spanned by the standard basis vector e1, and let X = {(L, v) ∈ Y : L = L0}. Then X is invariant under H := {g ∈ GLnF : gL0 = L0}. Now let’s check conditions (i) and (ii) of the recognition principle above: (i) is clear, since by linear algebra GLnF acts transitively

10 on the lines. And if (gL0, gv) = (L0, w) then g ∈ H. So we have a canonical isomorphism =∼ of G-sets G ×H X −→ Y . To complete this description, one should describe how H acts on X. Identifying X with the vector space L0, it is the linear action that pulls back scalar × multiplication along the homomorphism H−→F taking a matrix A to a11, the upper left entry: A · (L0, v) = (L0, a11v). Examples of this type occur frequently in geometry and topology.

8 Semi-direct products and group extensions

8.1 Prelude on products One of the most commonly used techniques in mathematics—and here I admit to stating the obvious, but bear with me—is to combine simple objects in some way to build complicated objects, and conversely to understand complicated objects by breaking them down into simpler objects. In the case of groups, for example, given groups H,K we can form a new group by taking the product H × K. Conversely, if we can decompose a given group G as a product G ∼= H ×K, where H and K are already understood, then—at least in principle—we can understand G. To carry out this latter strategy we need a recognition principle for such products. The reader has probably already seen this, but here is a reminder of how it works for an arbitrary finite number of factors:

Q Proposition 8.1 Suppose H1, ..., Hn are normal subgroups of G such that (i) Hi∩( j6=i Hj) = {e} and (ii) G = H1...Hn. Then the multiplication map m : H1 × H2 × ... × Hn−→G is an isomorphism of groups.

Proof: First note that m is a group homomorphism: For this, one needs to know that for i 6= j the elements of Hi,Hj commute with one another. But if x ∈ Hi and y ∈ Hj, then −1 −1 by normality xyx y ∈ Hi ∩ Hj = {e}. Then m is injective by (i) and surjective by (ii), completing the proof.

8.2 Semi-direct products But only if we are lucky will G decompose as a product. The next best thing is a “semi- ”, which we know describe. Suppose given groups H,K and a homomorphism ρ : K−→Aut H. Equivalently, we are given a left action of K on H by group automorphisms. Associated to this data we have the semi-direct product H oρ K: As a set it is just H × K, but with group multiplication defined by

(h1, k1) · (h2, k2) = (h1(ρ(k1)(h2)), k1k2). The proof that this multiplication defines a group structure is left as an exercise. Note that (h1, e) · (e, k2) = (h1, k2). So there is no harm in dropping the parentheses and writing hk in place of (h, k). Note also that the outer two factors in the displayed product just go along for the ride; all the action takes place with the inner two. Thus the essence of the multiplication rule is that it tells you how to commute k with h: kh = (ρ(k)(h))k. But we

11 already know how to do this, in any group: kh = (khk−1)k. The upshot of this discussion is that in the semi-direct product, the automorphism ρ(k) of H is the same thing as left conjugation by k. Note that by construction H is normal in H oρ K, and that H oρ K is the direct product H × K if and only if ρ is trivial if and only if K is normal (check this!). If ρ is understood, we often omit it from the notation and simply write H o K. Needless to say, abusive notation of this kind must be used with care.

Example. Suppose we ask: Are there non-abelian groups of order 21? With semi-direct products in hand, it is easy to construct such groups explicitly: Aut C7 is cyclic of order 6, so we can choose an injective homomorphism ρ : C3−→Aut C7 (there are two such ho- momorphisms) and form C7 oρ C3. More generally, given primes p, q with p|q − 1, we get non-abelian groups of order pq this way. Indeed one can show that every group of order pq has this form.

More importantly, many groups occuring “in nature” can be recognized as semi-direct products and thereby better understood. Here is a simple recognition principle:

Proposition 8.2 Suppose G contains subgroups H,K such that: (a) H is normal in G. (b) H ∩ K = e. (c) HK = G. −1 Then G = H oρ K, where ρ(k)(h) = khk .

Proof: Conditions (b),(c) imply that multiplication H × K−→G is a bijection. Since H is normal, the stated homomorphism ρ is defined, and a trivial check (which we have in effect already done above) shows that the multiplication on G is exactly the semi-direct product multiplication.

With this criterion in hand, you soon realize that semi-direct products are everywhere. Here are a few important examples, with details and verifications left to the reader:

Examples. 1. The dihedral group of order 2n is a semi-direct product Cn o C2, where C2 acts on Cn as multiplication by −1. 2. Sn = An o C2. 3. Let V be a finite dimensional vector space over a field F . The affine group Aff(V ) is defined to by the subgroup of P erm V generated by GL(V ) and the subgroup of translations Tv : x 7→ x + v . The group of translations is isomorphic to the additive group of V , and Aff(V ) = V o GL(V ). Here the action of GL(V ) on V by conjugation is the same as the n n standard action. When V = F , we also write AffnF in place of Aff(F ). 4. Consider the symmetric group Sp and the p-Sylow subgroup Cp generated by the

standard p-cycle (12...p). Then the normalizer NSp Cp is isomorphic to Aff(Fp) and hence is a semi-direct product of the form Cp o Cp−1. 5. BnF = UnF o DnF . What is the action of DnF on UnF ? × n 6. NnF = DnF o Sn, where Sn acts on DnF = (F ) by permuting the factors. This is a type of semi-direct product known as a , disussed further in the exercises.

12 8.3 Group extensions A consists of group homomorphisms

H −→i G −→π K, where π is surjective and i is an isomorphism onto the kernel of π. Thus without loss of generality we can, if desired, assume that i is just an inclusion and K = G/H. In fact we will often treat H as a subgroup and omit i from the notation. Note that H and K alone do not determine G, even when G is abelian. For example, if we are given a group extension C2−→G−→C2, then since |G| = 4 we know G is abelian, but without further information there is no way to know whether G is C4 or C2 × C2. The extension is central if H ⊂ C(G). Note, for example, that if H = C2 then the extension is automatically central. This leads to another example of the ambiguity inherent in group extensions: If we have an extension C2−→G−→C2×C2, then it is a central extension 3 but G could be any of the five groups of order 8 except C8:(C2) , C2 × C4, the dihedral group D8, or the group Q8. Each of these four groups fits into such an extension, as you can easily check. Note that a semi-direct product G := H o K fits into an extension H−→G−→K. In fact we can characterize the semi-direct products in terms of extensions. A group extension i π H −→ G −→ K splits if there is a homomorphism s : K−→G such that π ◦ s = IdK . We call s a splitting (or sometimes a “section”) of π.

Proposition 8.3 If G := H oK is a semi-direct product, the extension H−→G−→K splits. ∼ Conversely if H−→G−→K is a split group extension, then G = H oK with K acting on H by conjugation. More precisely, if s : K−→G is a splitting, K acts on H by k ·h = s(k)hs(k)−1.

Proof: If G = H o K, define s by s(k) = (e, k). Conversely if H−→G−→K is split, choose a splitting s : K−→G. Then the pair H, s(K) satisfies the recognition principle for semi-direct products (an easy check).

Remark. Note that to give a splitting s : K−→G is the same thing as giving a subgroup K0 ⊂ G such that π : G−→K maps K0 isomorphically to K.

Example. Let G be a group of order pq, where p, q are primes with p < q. I claim that G is a semi-direct product of the form Cq o Cp. First of all, by the third Sylow theorem there is a unique and hence normal q-Sylow subgroup, cyclic of order q. Hence there is an extension π Cq−→G −→ Cp. Now choose a p-Sylow subgroup H. Then π|H is injective, and hence an isomorphism. This proves the claim. Note that Cp acts on Cq by some homomorphism ∼ Cp−→Aut Cq = Cq−1, and hence for the action to be non-trivial we must have p|(q − 1) or equivalently q = 1 mod p. This fits with the third Sylow theorem because if H is not normal, then there are q p-Sylow subgroups.

9 Solvable and nilpotent groups

In attempting to analyze a group G by fitting it into an extension H−→G−→K—or equiva- lently, finding a normal subgroup H with quotient K—one simple possibility we might hope

13 for is to find such an extension with both H and K abelian. Or if we are feeling especially lucky, we might hope that in addition the extension is central. A more reasonable although still optimistic hope is that we can build G by a finite iteration of such extensions. This leads to the concepts solvable and nilpotent group in the respective cases.

Example. We can build S4 in two steps out of abelian groups. First we form the extension 2 C2 −→A4−→C3. Then we form the extension A4−→S4−→C2.

But we need a smoother way to think about “ up from extensions”. This is the subject of the next section.

9.1 Don’t fight it, filter it! An increasing filtration of a group G consists of subgroups

{e} ⊂ G1 ⊂ G2 ⊂ G3.... A decreasing filtration likewise consists of subgroups

G ⊃ G1 ⊃ G2 ⊃ G3 ⊃ ...

In either case the filtration stabilizes at Gn if Gk = Gn for all k ≥ n. An increasing (resp. decreasing) filtration is finite if Gn = G for some n (resp. Gn = {e} for some n). For finite filtrations there is no real difference between the increasing and decreasing cases, since one could always reverse the ordering to convert from one to the other. In fact this definition makes sense for any kind of object with subobjects: rings and subrings, vector spaces and sub-vector spaces, topological spaces and subspaces, etc. In group theory filtrations are classically known as “series”. I prefer the term “filtration” because it has a verb, “to filter”, that goes with it, and because it is the more widely used term across many different categories. But I will freely use both terms, just so you get used to them.

Example. Let p be a prime. Any A has a natural decreasing “p-adic” filtration 2 A ⊃ pA ⊃ p A ⊃ ... (which need not terminate; think of A = Z), as well as a natural p-torsion 2 filtration A[p] ⊂ A[p ] ⊂ ... (which need not terminate; think of A = Q/Z). Incidentally, it isn’t necessary for p to be a prime here, but the prime case is by far the most important.

Example. Note that according to our definition, a filtration of a finite group need not be a finite filtration. For example, suppose p, q are distinct primes, and G is a finite abelian q-group. Then pG = G and hence the p-adic filtration takes the form G ⊂ G ⊂ G ⊂ ...; it never reaches {e}. Similarly the p-torsion filtration takes the form {e} ⊂ {e} ⊂ ...; it never reaches G. Thus a finite filtration is not merely one with a finite number of distinct terms, but one that begins at the trivial subgroup and ends at the whole group (or vice-versa).

Usually the point of filtering a group (or anything else) is to arrange it in such a way that the quotient objects Gk/Gk−1 (increasing case) or Gk/Gk+1 (decreasing case) have some simple form that we understand; then the hope is that we can recover, perhaps by an

14 induction argument, information about G itself. This is the meaning of the motto: Don’t fight it, filter it! In our category of groups, as it stands the quotients are in general only sets, so we make the definitions:

Definition. An increasing filtration is subnormal if Gi is normal in Gi+1 for all i, and normal if Gi is normal in G for all i. Thus normal implies subnormal but not conversely (for a minimal counterexample to the converse, look at our old friend A4). Subnormal and normal decreasing filtrations are defined similarly. The classical terminology is “subnormal/normal series”.

It will be handy to have a term for the following simple construction: Suppose H−→G −→π K is a group extension, and we are given finite filtrations (which we can assume are increas- ing) of H and K. Then we get a filtration of G by “splicing” them together:

−1 −1 {e} = H0 ⊂ H1 ⊂ ... ⊂ H ⊂ π K1 ⊂ ...π Kn = G. We call this the splice of the two filtrations. Note that the splice of subnormal filtrations is subnormal, and that the list of quotient groups obtained is just the union of the quotient groups of the original filtrations. Caution: The splice of normal filtrations need not be normal. Once again, A4 provides a counterexample. In order for the splice to be a normal filtration, the filtration on H would have to be invariant under the conjugation action of K on the set of normal subgroups of H.

Now, let’s get on to our main examples.

9.2 Solvable groups Abelian groups are easier to understand than general groups. In the spirit of our filtration motto, therefore, it is reasonable to make the following definition: A group G is solvable if it admits a finite subnormal filtration with abelian quotients. (The terminology comes from Galois theory, where group theory originated.) We call such a filtration a solvable filtration (or series) for G. Any abelian group is solvable, and if H−→G−→K is a group extension with H,K abelian, then G is solvable. Before giving further examples, we establish some basic properties of solvable groups. First of all, there is a “functorial” decreasing filtration with abelian quotients of any group G, defined recursively by G0 = G and Gi+1 = [Gi,Gi]. We call this the filtration. Note that it is not only a normal filtration, but even a characteristic filtration; i.e. the subgroups in question are characteristic subgroups (invariant under arbitrary automorphisms of G). The first thing to note is that it has limited applicability; indeed for many groups it is a useless filtration: For example, if G is a nonabelian then it is the filtration G ⊃ G ⊃ G.... But if it does yield a finite filtration, i.e. one that reaches {e}, then G is solvable. We will prove the converse shortly.

Remark. By “functorial” we mean that any group homomorphism preserves the filtration. To make this fit precisely into the framework of category theory, we would define a category of filtered groups, with morphisms the filtration preserving homomorphisms, so that assigning

15 to G its commutator filtration is a functor to this new category. But there’s no compelling reason to do this at the moment.

Proposition 9.1 The following are equivalent: a) The commutator filtration of G is finite, i.e. G(m) = {e} for some m. b) G admits a finite normal filtration with abelian quotients. c) G is solvable.

Proof: Clearly (a) ⇒ (b) ⇒ (c). Now suppose G is solvable, and let G = H0 ⊃ H1 ⊃ ... ⊃ Hm = {e} be a solvable filtration. Then G/H1 is abelian, so [G, G] ⊂ H1. Similarly, since H1/H2 is abelian, we have G(2) ⊂ [H1,H1] ⊂ H2. Continuing in this manner, we find that G(k) ⊂ Hk for all k. Hence G(m) = {e}, proving that (c) ⇒ (a).

Proposition 9.2 Solvable groups are closed under taking subgroups, quotients, extensions, and finite products.

Proof: We first show that if G is solvable, so is any quotient group. Suppose π : G−→H is a surjective homomorphism. Let G ⊃ G1 ⊃ G2 ⊃ ... be a solvable filtration for G, and consider the filtration π(Gi) of H. It is a finite subnormal filtration, and the quotients are abelian because π(Gi)/π(Gi+1) is a quotient of Gi/Gi+1. Hence H is solvable. The case of subgroups is equally straightforward, and left to the reader. Next we show that solvable groups are closed under extensions; that is, if H−→G−→K is a group extension and H,K are solvable, so is G. This is immediate because given solvable filtrations for H,K we can splice them to get a solvable filtration for G (recall that subnormality is preserved under splicing, and the quotients remain abelian because they are in fact identical to the original quotients). Finally, consider products. The product of two solvable groups G, H is solvable because of the extension H−→G × H−→G. Induction on the number of factors then shows that any finite product of solvable groups is solvable.

Remark: It follows that the solvable groups can be described as the smallest class of groups that contains the abelian groups and is closed under extensions.

Now, here is one of the most important solvable groups. We could work over a general commutative , but to avoid distractions we will stick to the case of a field F . Recall that BnF ⊂ GLnF is the Borel subgroup of upper triangular matrices.

Proposition 9.3 BnF is solvable.

× Proof: Note that B1F = F , and that there is a surjective homomorphism π : BnF −→Bn−1F given by simply projecting A ∈ BnF onto its upper left (n−1)×(n−1) block. By induction we can assume Bn−1F is solvable, so it suffices to show Ker π is solvable. Now Ker π is the “right column group” consisting of matrices that equal the identity in the first n − 1 u × columns, which we will quaintly denote RCnF . It fits into an extension RCn −→RCn−→F , u where the second map is just projection on the nn coordinate and RCn is therefore the

16 u subgroup with ann = 1 (the superscript u if for “unipotent”). One easily checks that RCn n−1 is isomorphic to the additive group F , and in particular is abelian. So RCnF is solvable and the proof is complete.

See the exercises for a discussion of the commutator series of BnF .

The finite p-groups are another important family of solvable groups. However, they satisfy the even stronger property of “nilpotence”, as we will show in the next section.

There are a number of theorems showing that under certain restrictions on the prime factors of n, every group of order n is solvable. Here are three such theorems, in increasing order of difficulty (the first is by far the easiest, and is demoted to the status of “proposition”):

Proposition 9.4 a) If n = pq with p, q prime, then every group of order n is solvable. b) If n = pqr with p, q, r distinct primes, then every group of order n is solvable.

Proof: Exercise. Part (a) is trivial from things we’ve already proved. Part (b) is a little more interesting.

The next result is known as Burnside’s paqb theorem.

Theorem 9.5 Suppose p, q are primes and |G| = paqb. Then G is solvable.

The proof is a beautiful application of representation theory, as we will show later.

Corollary 9.6 Every group of order < 60 is solvable.

Proof: 60 = 22 · 3 · 5 is the smallest number that is neither the product of three distinct primes nor of the form paqb. (It’s also a nice exercise to prove the corollary directly, without Burnside’s theorem.)

The next theorem is due to Feit and Thompson in 1963.

Theorem 9.7 Every finite group of odd order is solvable.

The original proof is 200 pages long, and as far as I know has never been simplified.

9.3 Nilpotent groups

An increasing normal filtration {e} = G0 ⊂ G1 ⊂ ... of G is central if Gi+1/Gi ⊂ C(G/Gi). We say that G is nilpotent if admits a finite central filtration, i.e. one that terminates at G. Note that nilpotent groups are solvable, since any central filtration is a solvable filtration. On the other hand, S3 is solvable but not nilpotent.

Proposition 9.8 Every finite p-group is nilpotent.

17 Proof: Let G be a finite p-group. Then C(G) is non-trivial. By induction on order we can assume G/C(G) has a finite central filtration; splicing with the one-step filtration {e} ⊂ C(G) yields the result.

In fact any group has a functorial, normal central filtration, sometimes called the as- cending , defined recursively as follows: Let C1 = C(G). Having defined the normal subgroup Ck, let π : G−→G/Ck be the quotient homomorphism, and set −1 Ck+1 = π C(G/Ck). It is clear that Ck+1 is normal. The next proposition is analogous to Proposition 9.1.

Proposition 9.9 The following are equivalent: a) The ascending central series of G is finite; i.e., ends at G. b) G is nilpotent.

Proof: Clearly (a) ⇒ (b). To show that (b) ⇒ (a), assume given a central filtration Gi with Gm = G and show inductively that Gi ⊂ Ci. Then Cm = G, as desired.

Before proceeding further, it will be convenient to generalize the concept “nilpotent group” to nilpotent group action. Suppose K,G are groups and K acts on G by group automorphisms. We say that the action in nilpotent if G has a finite subnormal filtration {e} ⊂ G1... ⊂ Gm = G such that (i) each Gi is invariant under the K-action, and (ii) the induced K-action on each quotient Gi/Gi−1 is trivial. We call such a filtration K-nilpotent.

Example. Let G act on itself by conjugation. This action is nilpotent if and only if G is nilpotent.

Example. Consider a field F and take K to be the group of upper triangular unipotent n matrices UnF . For G we take the additive group F , with its standard left UnF action. Filter F n by the F i’s (where as always, our default inclusion F i ⊂ F n is in the first i coordinates). This is automatically a subnormal (indeed normal) filtration, since F n is abelian, and satisfies the nilpotent action conditions (i)-(ii) by definition of UnF . Note: From the point of view of this example, it would have made more sense to use the term “unipotent action” in place of “nilpotent action”. But such terminology conflicts are inevitable, and one just has to live with them.

Nilpotent groups are not closed under extensions (think of S3, for example). Our next definition compensates for this deficiency: A group extension H−→G−→K is nilpotent if the conjugation action of G on H is nilpotent. Any central extension is nilpotent, for example, or more generally any extension in which the conjugation action of G on H is trivial. The extension C3−→S3−→C2 is not nilpotent.

Proposition 9.10 The class of nilpotent groups is closed under subgroups, quotients, nilpo- tent extensions and finite products.

Proof: Let G be nilpotent, H a subgroup. Since the action of G on itself by conjugation is nilpotent, so is the restriction of this action to H, i.e. the action of H on G by conjugation.

18 But H is invariant under the latter action, and we get a finite central filtration of H by intersecting with a given such filtration for G. The case of quotients is also straightforward, and left to the reader. Now suppose H−→G−→K is a group extension with K nilpotent and G acting nilpo- tently on H by conjugation (which implies that H is nilpotent). Choose a G-nilpotent filtration of H and a nilpotent filtration of K; splicing these yields a nilpotent filtration of G, as desired. Details are left to the reader. If H,G are nilpotent, then H−→G × H−→G is clearly a nilpotent extension, since the conjugation action of G × H on H factors through H. So H × G is nilpotent. It then follows by induction on the number of factors that any finite product of nilpotent groups is nilpotent.

Next is one of the most important examples. Let F be a field, and recall that the unipotent group UnF is the group of upper triangular n×n matrices with 1’s on the diagonal. Since it is a subgroup of the BnF , it is solvable. But more is true:

Proposition 9.11 UnF is nilpotent.

Proof: We follow closely the proof already given for solvability of BnF . As in that case, there is a group extension

u RCn F −→UnF −→Un−1F, u where the second map is projection on the upper left (n − 1) × (n − 1) block and RCn F is again the “unipotent right column group”; for example RC3F consists of matrices

 1 0 a     0 1 b  0 0 1

By induction we can assume Un−1F is solvable, so it suffices to show that the action of UnF u u on RCn F by conjugation is nilpotent. Since RCn F is abelian, the action of UnF factors u through Un−1F , so what we need to show is that the conjugation action of Un−1F on RCn F u ∼ n−1 is nilpotent. But under the evident isomorphism RCn F = F (where the entries of the right column are ordered from top to bottom), this action corresponds to the standard linear n−1 action of Un−1F on F . We saw earlier that this latter action is nilpotent, so the proof is complete.

Here are two more interesting facts about nilpotent groups. Both are false for solvable groups; the reader can easily supply examples.

Proposition 9.12 Let G be a nilpotent group, H a normal subgroup. Then H ∩C(G) 6= {e}.

Proof: Since G acts nilpotently on itself by conjugation, and H is invariant, it acts nilpotently on H. In particular H has a non-trivial subgroup H1 on which G acts trivially, so H1 ⊂ H ∩ C(G).

19 Proposition 9.13 Let G be a nilpotent group, H a proper subgroup of G. Then H is a proper subgroup of its normalizer NGH.

Proof: If C := C(G) is not contained in H, then H 6= CH ⊂ NGH. If C ⊂ H, then by induction we can assume H/C 6= NG/C H/C; it follows at once that H 6= NGH.

Now we can neatly characterize finite nilpotent groups.

Theorem 9.14 A finite group G is nilpotent if and only if G is a product of p-groups (where p ranges over the prime divisors of |G|).

Proof: We have shown that any p-group is nilpotent. Since any finite product of nilpotent groups is nilpotent, this proves the “if”. Conversely, suppose G is nilpotent, let p divide |G|, and let P be a p-Sylow subgroup. Then NG(NGP ) = NGP (see the section on the Sylow theorems). By Proposition 9.13 this is impossible unless NGP = G, so P is normal. So if p1, ..., pm are the prime divisors of G, with corresponding unique pi-Sylow subgroups Pi, the natural multiplication map φ : P1 × ... × Pm−→G is an injective group homomorphism. Comparing orders, we see that it must be an isomorphism.

The descending central series of a group G is defined recursively by C0 = G and Ck+1 = [G, Ck]. Thus C1 = [G, G], so this starts off the same as the commutator series. But at the next step we take [G, [G, G]] instead of [[G, G], [G, G]]. By construction the descending central series is a characteristic (hence normal) filtration, whose quotients are not only abelian but satisfy Ck/Ck+1 central in G/Ck+1. Hence if the descending central series is finite, i.e. ends at {e}, reversing the order yields a finite central increasing filtration of G. This proves one direction of the following proposition; the converse is left to the reader.

Proposition 9.15 G is nilpotent if and only if the descending central series is finite, i.e. Cn = {e} for some n.

We conclude by mentioning a class of finite groups lying between the nilpotent and solvable groups. If G is a finite group, G is supersolvable if it has a finite normal filtration with cyclic quotients.

Proposition 9.16 For finite groups G, nilpotent ⇒ supersolvable ⇒ solvable.

The second implication is immediate, while the first is left as an exercise. Note that S3 is supersolvable but not nilpotent, while A4 is solvable but not supersolvable. One can also check that any solvable finite group admits a subnormal filtration with cyclic quotients; hence the insistence on normal filtrations in the definition of supersolvable groups is key. Supersolvable groups have nice representation-theoretic properties, as we will see in [Serre], §8.5.

20 10 Simple groups and perfect groups

10.1 Simple groups A group G is simple if it has no non-trivial proper normal subgroups. If G is abelian, then G is simple if and only if it is cyclic of prime order (an easy exercise). Non-abelian finite simple groups were not completely classified until 2004; the proofs occupy thousands of pages. The formidable Feit-Thompson theorem implies that every non-abelian finite simple group has even order, but this is just the first little step! Note that since every group of order < 60 is solvable, there are no non-abelian simple groups of order < 60 (indeed these two statements are equivalent).

Proposition 10.1 The An is simple for n ≥ 5.

For a short and interesting proof, see Artin’s undergraduate algebra text (proofs can be found also in [Hungerford] and [Dummit-Foote]). Another infinite family of simple groups is given by the projective special linear groups PSLnF , F a field. These are defined by PSLnF = SLnF/C, where C := C(SLnF ) is the center. Since C(GLnF ) consists of the scalar matrices, it is easy to see that C(SLnF ) is also just the scalar matrices with deter- minant 1, and hence is a finite of order dividing n. It turns out that with just two exceptions, PSLnF is simple for all n, F . We’ll prove this in a later section for n = 2, and sketch the proof for general n. By taking F to be a finite field we get an infinite family of finite simple groups. This is as far as we’ll go.

10.2 The Jordan-Holder theorem Let G be a group. A Jordan-Holder filtration of G is a finite increasing subnormal filtration with simple quotients. Such a filtration need not exist; for example, an abelian group admits a Jordan-Holder filtration if and only if it is finite (easy exercise). On the other hand, clearly any finite group admits a Jordan-Holder filtration. Here is the Jordan-Holder theorem:

Theorem 10.2 If G admits a Jordan-Holder filtration, then the list of simple groups occur- ing as the quotients is independent of the choice of filtration, up to ordering.

We will rarely—if ever—use this theorem, so we won’t prove it here. For a proof and further discussion see e.g. [Hungerford]. (We will, however, prove and use the analogous theorem for modules over a ring.)

10.3 Perfect groups

A group G is perfect if G = [G, G]. Thus G is perfect if and only if Gab = {e} if and only if every homomorphism from G to an abelian group is trivial. Some easy facts about perfect groups:

• A group is both perfect and solvable if and only if it is trivial.

21 • Any simple non-abelian group is perfect. • Perfect groups are closed under quotients, extensions, and arbitrary products (but not under subgroups). Note that a perfect group need not be simple. For example, any product of non-abelian simple groups such as A5 × A5 is perfect, but certainly not simple. We will see below that with two exceptions, SLnF is perfect (but typically has non-trivial center, hence need not be simple). Indeed for us this is the main reason to bother introducing the concept of a perfect group; we will use it as a stepping stone toward proving PSLnF is (almost always) simple.

11 On SLnF and PSLnF

Recall (see “Preliminary Remarks”) that SLnF is the group of n × n matrices over the field F with 1. Note that GLnF fits into a split extension

× SLnF −→GLnF −→F .

Many basic subgroups of SLnF are defined by simply intersecting with the corresponding subgroups of GLnF . For example, define the Borel subgroup of SLnF by SBnF = BnF ∩ SLnF , the upper triangular matrices with determinant 1. Although one rapidly grows tired of putting the “S” in SBnF and its cousins, we’ll do so for a little while. Thus SDnF denotes diagonal matrices of determinant 1, SUnF = UnF (since the unipotent subgroup already consists of determinant 1 matrices), and SNnF = NnF ∩ SLnF . Here one can check

that SNnF = NSLnF SDnF —provided, as usual, that F 6= F2. The one significant difference to watch out for concerns the Weyl group. One might think that we should just take matrices of determinant 1, but this turns out to be the wrong thing to do. The way to think of it is as follows: In GLnF we have the extension

DnF −→NnF −→WnF, ∼ where WnF = Sn and a splitting of the extension is already given by using permutation matrices. If F 6= F2 there is an analogous extension

SDnF −→SNnF −→SWnF,

where SWnF is by definition SNnF/SDnF and is again isomorphic to the symmetric group Sn. But unless char F = 2, the extension doesn’t split: the problem is that transpositions (as permutation matrices) have determinant −1 and there is no way to lift them to elements of order 2 in SNnF . It’s already a problem when n = 2, as you can check. When char F 6= 2, they do however lift to elements of order 4; e.g. for n = 2 use 0 −1 ! 1 0 or its inverse. The upshot of this discussion is that we just have to live with the fact that the extension SDnF −→SNnF −→SWnF = Sn usually doesn’t split. In practice, fortunately,

22 this does not cause any difficulties. To illustrate, let’s first clarify our definition of SWnF : If char F = 2, it is just the permutation matrices as before, i.e. is equal to WnF . If char F 6= 2, ∼ ∼ SWnF = SNnF/SDnF . In all cases SWnF = WnF = Sn. Now consider the SLn version of the Bruhat decomposition.

Theorem 11.1 a SLnF = SBnF wSB˜ nF, w∈WnF

where in the case char F 6= 2, w˜ is any preimage of w in SNnF (in the characteristic 2 case it is just a permutation matrix as usual).

Proof/Discussion: First of all, note that the expression on the right is well-defined. Any two choices ofw ˜ differ by an element of SDnF ⊂ SBnF , so the double SBnF wSB˜ nF is independent of the choice. The theorem then follows easily from its GLn analogue; no new fussing about with row reduction is needed. Certainly the indicated double cosets are distinct and hence disjoint, by comparing with the GLn result. And if g ∈ SLnF , we can −1 write g = b1wb˜ 2 with b1, b2 ∈ BnF by the GLn result, with det b2 = (det b1) . We can write 0 0 b1 = b1d with b1 ∈ SBnF and d ∈ DnF . Then sincew ˜ normalizes DnF , we have

0 0 g = b1w˜(d b2), 0 0 where det d = det d = det b1 and hence d b2 ∈ SBnF .

Carrying around the decorations S, n, F gets tiresome, as does thew ˜ notation. So as long as we have firmly declared that our context is SLnF , we prefer to abbreviate the above Bruhat decomposition in a slightly abusive but much more pleasant notation as

a SLnF = BwB. w∈W

Thus it is understood that B means SBnF , and so on.

11.1 On SL2F

The goal of this section is to study the structure of SL2F in detail, and in particular to prove:

Theorem 11.2 If F is a field, then PSL2F is a simple group except when F = F2 or F = F3. ∼ ∼ In the exercises you show that PSL2F2 = S3 and PSL2F3 = A4, so these two cases are not simple. Taking F = Fq to be a finite field with q > 3, the theorem gives us a large, potentially new class of simple groups. I say “potentially new” because it might happen that some of these groups are isomorphic to alternating groups. Indeed |PSL2F5| = 60, and ∼ hence PSL2F5 = A5 (see the exercise on simple groups of order 60). It turns out that this is the only such coincidence, however, a fact that one can check by hand for small q by just

23 checking the orders are not of the form n!/2. For example, PSL2F7 is simple of order 168 and so is not an alternating group. The group SL2F is important for much more general reasons, as it is the most basic example of a “semi-simple ”, or in the case F = R, C of a “semi-simple ”. It plays a central role in representation theory and Lie theory, as well as in certain parts of algebraic topology, algebraic geometry, and combinatorics.

11.1.1 Some handy formulas in SL2

We will need some elementary formulas in SL2F . They are all easy to check directly. First some notation: If t ∈ F we set x(t) =

1 t ! 0 1

and y(t) =

1 0 ! t 1

If t ∈ F × we set w(t) =

0 t ! −t−1 0

and h(t) =

t 0 ! 0 t−1

These satisfy:

1. w(1)x(t)w(1)−1 = y(−t)

2. h(t) = w(t)w(1)−1

3. w(t) = x(t)y(−t−1)x(t)

4. [h(s), x(t)] = x((s2 − 1)t)

5. h(s)x(t)h(s)−1 = x(s2t)

6. x(t)h(s)x(t)−1 = h(s−1)x((s2 − 1)t)

Note that any of the last three formulas easily determines the other two, and that there are analogues with x(t) replaced by y(t). Let w = w(1).

24 11.1.2 Properties of SL2F

Note that the center C := C(SL2F ) is trivial if char F = 2 and is C2 (i.e. ±Id) otherwise. ` The Bruhat decomposition SL2F = B BwB immediately implies:

Proposition 11.3 B is a maximal proper subgroup of SL2F .

− Proposition 11.4 SL2F is generated by the subgroups U, U (together), i.e. by the elements x(t), y(t), t ∈ F .

Proof: Let K denote the subgroup generated by U, U −. By the Bruhat decomposition we know that SL2F is generated by U, D, and w; that is, by the elements x(t), h(t), w(1). Since w(1) ∈ K by Formula (3), and hence h(t) ∈ K by Formula (2), we are done.

Proposition 11.5 SL2F is perfect if F 6= F2, F3.

Proof: By the preceeding proposition, it suffices to show that x(a), y(a) are for all a ∈ F . By Formula (1), we need only consider x(a). By Formula (4) we see that x(a) is a commutator provided there is an s ∈ F × with s2 6= 1, and this latter statement is true if and only if |F | > 3.

The two exceptional cases are not perfect (see the exercises).

11.1.3 Simplicity of PSL2 We restate the theorem:

Theorem 11.6 If F is a field, then PSL2F is a simple group except when F = F2 or F = F3.

Proof: Assume |F | > 3. We abbreviate G := SL2F . It is equivalent to show that if H is a normal subgroup of G, then either H ⊂ C (the center) or H = G. Since H is normal, HB is a subgroup of G, and as it contains B, by maximality of B we have either HB = B or HB = G. Suppose HB = B. Then H ⊂ B, and as H is normal and wBw−1 = B−, we have H ⊂ B ∩ B− = D. But by Formula (6), any subgroup of D that is normal in G must consist of elements h(s) with s2 = 1, i.e. must lie in C. Now suppose HB = G. Then G/H ∼= B/(B ∩ H). Since perfect groups and solvable groups are preserved under quotients, we conclude using Proposition 11.5 (and the assump- tion |F | > 3) that G/H is both perfect and solvable, hence trivial. So H = G, as desired.

25 11.2 Simplicity of PSLnF : a sketch

In this section we sketch how the results on SL2 extend to SLn. In particular we sketch a proof that PSLnF is simple for all n ≥ 3 and all F . We use the following notation: For each i, 1 ≤ i < n, we let Gi ⊂ GLnF denote the block diagonal subgroup consisting of just one 2 × 2 SL2F block in the i, i + 1 . For example, if n = 4 then G2 is the subgroup

 1 0 0 0   0 a b 0       0 c d 0  0 0 0 1 with ad − bc = 1. In each such block we have copies of the elements x(t), y(t), h(t), w(t) defined above for SL2F , which we denote xi(t), yi(t), etc. Similarly we have subgroups Ui,Bi,Di,Ni etc. of Gi corresponding to U, B, D, N in SL2F . Caution: This conflicts with our earlier notation,in which for example Bi denoted the upper triangular matrices in GLiF . In the displayed matrix above, B2 is the subgroup defined by c = 0.

Proposition 11.7 SLnF is generated by the subgroups Gi, and hence is generated by the elements xi(t), yi(t), t ∈ F .

Proof: The first statement follows by the usual row/column reduction (or Bruhat decomposi- tion) argument, since all the elementary row/column operations can be realized by repeated left/right multiplication by elements of the Gi’s. The second statement then follows by what we proved for SL2.

Proposition 11.8 SLnF is perfect except when n = 2, F = F2, F3.

Proof: The case n = 2 was proved earlier. If F 6= F2, F3, the general case follows immediately from the fact that the Gi’s are perfect together with the previous proposition. However, one can give a uniform proof for all F and n ≥ 3 as follows: It suffices to show each xi(t) is a commutator. These are all conjugate in SLnF (via a permutation of coordinates), so we can assume i = 1, in which case it is enough to prove the result for n = 3. But x1(t) is conjugate to x13(t), i.e.

 1 0 t     0 1 0  0 0 1

Since [x1(a), x2(b)] = x13(ab), x13(t) is a commutator for all t and we’re done.

Now the other key ingredient in the PSL2 simplicity proof was that the Borel subgroup B ⊂ PSL2F is a maximal subgroup. This is clearly not true for higher n, as there are various block-triangular groups that contain B. Explicitly if a = (a1, ..., ar) is an ordered partition of n, there is an associated parabolic subgroup Pa consisting of block triangular

26 matrices whose diagonal blocks have size a1, ..., ar. For example if n = 6 and a = (3, 1, 2), then Pa consists of matrices  a b c ∗ ∗ ∗     d e f ∗ ∗ ∗     g h i ∗ ∗ ∗     0 0 0 j ∗ ∗       0 0 0 0 k l  0 0 0 0 p q where the ∗ entries are arbitrary and the three diagonal blocks have the product of their equal to 1. Note that B itself corresponds to the partition (1, 1, ..., 1), while SLnF corresponds to (n). Note also that Pa is generated by B together with the wi’s it contains. The total number of parabolic subgroups is thus 2n−1. The surprising fact is:

Proposition 11.9 Suppose H is a subgroup of SLnF containing B. Then H = Pa for some a.

In particular there are only finitely many subgroups containing B. Note that when n = 2 the proposition says that B is a maximal proper subgroup.

Theorem 11.10 PSLnF is simple unless n = 2 and F = F2, F3.

Proof: The case n = 2 was proved earlier, so assume n ≥ 3 and let H ⊂ SLnF be a normal subgroup. Then HB is a subgroup containing B, so HB = Pa for some partition a. If Pa = B then H ⊂ B, and as in the case n = 2 we find that H ⊂ D, from which it follows that in fact H ⊂ C, C being the center (i.e. scalar matrices of determinant 1). The reader can fill in the details of this step. If Pa = SLnF , then as in the case n = 2 we find that SLnF/H is both perfect and solvable, hence trivial, so H = SLnF and we’re done. The new step that remains is to show that the case Pa 6= B,SLnF can’t occur. We sketch briefly the ideas involved. Suppose Pa 6= B,SLnF . Then Pa contains some but not all of the wi’s. Hence there must exist i, i + 1 such that wi ∈ Pa and wi+1 ∈/ Pa. Since BwiB ⊂ Pa = HB, we have (BwiB) ∩ H 6= ∅. Since H is normal, it follows that −1 (wi+1BwiBwi+1) ∩ H 6= ∅. From this one can show that wi+1wiwi+1 ∈ HB = Pa. But this is false, as one can check from the definition of Pa. So we have a contradiction and the proof is complete.

Taking F to be a finite field, this yields a large family of finite simple groups. With a very small number of exceptions, they are distinct from each other and from the alternating groups (up to isomorphism). In fact if memory serves, the only (?) exception besides the ∼ ∼ coincidence PSL2F5 = A5 already mentioned is PSL2F7 = GL3F2. Note these last two groups have order 168. To get the isomorphism one can show that in fact there is only one simple group of order 168 up to isomorphism; for an elaborate proof of this see [Dummit- Foote].

27 12 Exercises

Note: Selected exercises will be assigned. Remember too that your mission is not merely to find any old proof; always strive for a simple, elegant argument, and of course make full use of the machinery that we develop. Needless to say, a “simple, elegant” argument may or may not come to mind, or even be possible, but it should always be your goal. Notice. The Feit-Thompson theorem is off-limits unless explicitly allowed!

A. G-actions.

A1. By Cayley’s theorem (which is trivial, from a modern perspective), every finite group is isomorphic to a subgroup of Sn for some n, where the n provided by the proof is n = |G|. On the other hand, G may well be isomorphic to a subgroup of Sm for some much smaller m (think of Sm itself, for example). Show, however, that for G = Q8 (the quaternion group of order 8), n = 8 is the minimal n for which the conclusion of Cayley’s theorem holds.

A2. Prove by first finding a suitable set on which the group in question acts: a) GL2F2 is isomorphic to S3, and Aff2F2 is isomorphic to S4. b) PSL2F3 is isomorphic to A4. c) If G is a simple group of order 60, then G is isomorphic to A5.

B. Wreath products.

R Let H be a group, G ⊂ Sn a subgroup. The wreath product H G is the semi-direct n n product H o G, where G acts on the left of H by permuting the factors. Thus the elements R of H G have the form (h1, ..., hn)g, with multiplication determined by the formula

g · (h1, ..., hn) = (hg−11, ..., hg−1n)g. Note that the notation is defective, since H R G depends not just on G but on the particular n and the way G is embedded in Sn. But this should cause no confusion in context. In R n particular the notation H Sn always means H o Sn unless otherwise specified.

B1. Partition [mn] into n blocks (subsets) of equal size m, compatibly with the standard order on [mn]. Let Γ ⊂ Smn denote the subgroup of all block-preserving ; by this we mean that γ ∈ Γ is allowed to permute the blocks as well as the elements within a ∼ R particular block. Then Γ = Sm Sn.

R B2. Suppose G ⊂ Sm and K ⊂ Sn. Regard Sm Sn as a subgroup of Smn as in 2.1. Then there is an isomorphism Z Z Z Z (H G) K ∼= H (G K). R R R Consequently expressions such as H G1 G2... Gr are unambiguous (assuming Gi is

given as a subgroup of Sni ). In particular we can define the r-fold iterated wreath product R r H of a subgroup H ⊂ Sn. It is a subgroup of Snr .

28 B3. Wreath products at Wimbledon. The Wimbledon tennis tournament begins with a “draw” of 128 players. Player no. 1 plays no. 2, no. 3 plays no. 4, the winner of 1-2 plays the winner of 3-4, and so on. Some care is needed in determining the draw; for instance the two (theoretically) best players should be placed in opposite halves, say one at position 1 and the other at position 128, so that if they meet at all it will be in the finals. The iterated R 7 wreath product C2 ⊂ S128 can be thought of as the group of permutations of the draw leaving it “essentially unchanged”, and the number of “essentially distinct” possible draws is (128!)/2127. Explain.

m B4. Sylow subgroups of symmetric groups. Let n = a0 + a1p + ... + amp be the p-adic expansion of n (so 0 ≤ ai < p). Then Sn has a p-Sylow subgroup P with

m Z i ∼ Y ai P = ( Cp) . i=1 R i Here Cp is the i-th iterated wreath product of Cp, and for each i we are taking the product of ai copies of it, 1 ≤ i ≤ m.

C. Sylow subgroups of general linear groups of finite fields. In this exercise p is a prime and d q = p for some d, while Fq denotes a field with q elements. For any prime `, ν`n denotes the exponent of ` in the prime factorization of n.

n (2) Qn i C1. Answer/show: a) |GLnFq| = q i=1(q − 1) b) The unipotent subgroup U is a p-Sylow subgroup. c) How many p-Sylow subgroups are there? d) What is the order of SLnFq? Order of PSLnFq? × × Note: Recall that Fp is a cyclic group. This is true for any finite field, so Fq is cyclic of order q − 1. We haven’t proved the general case yet, but you can assume it if necessary in part (d).

Note: Some of the remaining C problems may require a little knowledge of finite fields beyond what has been discussed in class, and therefore may be postponed.

C2. Now let ` be a prime 6= p. Assume that ` divides q−1, and that if ` = 2 then 4|(q−1).

Then DW = NGLnFq D contains an `-Sylow subgroup of GLnFq. Hence if ν`(q−1) = a, GLnFq R has an `-Sylow subgroup isomorphic to C`a L, where L is an `-Sylow subgroup of Sn (cf. 3.1). Remarks: (i). If ` is odd and ` doesn’t divide q − 1, the `-Sylow subgroups are of a similar nature but the details are more complicated. To pursue this point, think about the extension of Fq obtained by adjoining an `-th root of unity. (ii). Why the restriction when ` = 2? One of my favorite mottos is the doubly nonsensical “4 is an odd prime”. The key, and elementary, number-theoretic fact behind this motto is n the following: Suppose ν`(x − 1) = a ≥ 1. Then ν`(x − 1) = a + ν`n provided that either ` is odd or a ≥ 2. When ` = 2 and a = 1 this fails, e.g. for x = 3, n = 2.

2 C3. Let a = ν2(q − 1).

29 a) If q = 3 mod 4, the 2-Sylow subgroups of GL2Fq are isomorphic to the semi-dihedral a+1 group SDa+1 of order 2 , defined for a ≥ 3 as follows: Let C2a , C2 have generators x, y 2a−1 respectively, and let C2 act on C2a by y · x = x . Then SDa+1 = C2a o C2. 2 Suggestion: Identify Fq with Fq2 and consider the group of units of Fq2 together with the Frobenius a 7→ aq.

R b) If q = 3 mod 4 and n = 2m, then GL2Fq Sm (embedded in GLnFq in the evident way) ∼ R contains a 2-Sylow subgroup P . Hence P = SDa+1 Q, where Q is a 2-Sylow subgroup of Sm. What happens for n odd?

c) If q is any odd prime power, the 2-Sylow subgroups of SL2Fq are isomorphic to the a generalized quaternion group Qa of order 2 .(Qa is defined as follows for a ≥ 3: Write the + a−1 as H = C ⊕ Cj. Then Qa is the subgroup of H generated by j and the 2 -st roots of unity in C.) D. Balanced products and induced G-sets.

D1. Let F be a field, and let Xn ⊂ MnF denote the subset consisting of matrices with n distinct eigenvalues, all of which lie in F . In this problem you will give two alternate ways of × n thinking about Xn. Let Zn = Dn ∩Xn, which we identify with the subset of (F ) consisting of n-tuples with no repeated entries. n P n a) Let Yn denote the set of ordered n-tuples (L1, ..., Ln) of lines in F such that Li = F . ∼ Then there is a “natural” bijection Zn ×Sn Yn = Xn. (Here “natural” is meant informally, i.e. the definition of the bijection should flow naturally out of the given data.)

b) Note that GLnF acts on Xn by conjugation, and Zn is invariant under the restriction of this action to NnF . Use the recognition principle for induced G-sets to show that the canonical map GLnF ×NnF Zn−→Xn is a bijection.

Remark: For fans of topology, I note that for F = R, C, the set Xn is a subspace of MnF (with its usual topology). The other two spaces in (a), (b) have quotient making the above bijections homeomorphisms.

D2. Let H be a subgroup of G and let X be a G-set. Then there is a natural isomorphism ∼ of G-sets G ×H X = (G/H) × X, where the target has the product G-action. In particular, the induced action g1 · (g, x) = (g1g, x) on G × X is isomorphic to to the product action.

E. Maximal and minimal subgroups of nilpotent and solvable groups. “Maximal subgroup” means “maximal proper subgroup”, while “minimal subgroup” means “minimal non-trivial subgroup”. Here are four interesting facts to prove; G is al- ways a finite group.

E1. Suppose G is solvable. Then a) Every minimal normal subgroup is an elementary abelian p-group. (A finite abelian p-group A is elementary if every element of A has order p, or equivalently A is a product of of Cp’s.) b) Every maximal subgroup has prime power index.

30 E2. Suppose G is nilpotent. Then: a) Every minimal normal subgroup is central of prime order. b) Every maximal subgroup is normal and has prime index.

E3. The converse of 2b holds in the form: If G is a finite group such every maximal subgroup is normal, then G is nilpotent. (Suggestion: Show that every p-Sylow subgroup is normal.)

F. Properties of unipotent and Borel groups.

1. UnF is generated by the subgroups Ui,i+1 for 1 ≤ i < n. 2. C(UnF ) = U1,n. 3. [UnF,UnF ] is the subgroup of elements whose (i, i + 1) entries are zero, 1 ≤ i < n. 4. Determine the ascending central, descending central and commutator series for U4F . (Or if feeling ambitious, do it for general n.)

Structure of BnF . 1. If F 6= F2,[BnF,BnF ] = UnF . Hence for all fields the commutator series of BnF is determined by that of UnF . 2. Determine the upper and lower central series of BnF for F 6= F2. (They won’t get far!)

M. Miscellaneous problems. These are mostly problems already suggested in the body of the notes.

M1. Show that if either n = pq with p, q prime, or n = pqr with p, q, r distinct primes, then every group of order n is solvable.

M2. Without using Burnside’s paqb theorem, show that every group of order < 60 is solvable.

M3. Show that every finite nilpotent group is supersolvable.

M4. Show that an abelian group admits a Jordan-Holder filtration if and only if it is finite. ∼ M5. Let P be a p-Sylow subgroup of Sp (p prime). Show that NSp P = Aff1Fp.

31