1. Group actions and other topics in group theory

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1 1. Group actions and other topics in group theory October 11, 2014 The main topics considered here are group actions, the Sylow theorems, 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 set of the first n natural numbers 1, 2,..., n. P k [n] denotes the set of k-element subsets of [n]. 1 Group actions 1.1 Definition of a group action 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 left and right 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 up to this point, we haven t even used the existence of inverses, so exactly the same definition makes sense for left and right monoid 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-equivariant map if φ(gx) = gφ(x) for all g G, x X. Then G-sets and G-maps form a category that we will denote 1

2 G-Set. The fixed-point set X G is defined by X G = {x X : gx = x g G}. This defines a functor G-set Set in the evident way. The isotropy group G x of a point x X is defined by G x = {g G : gx = x}; clearly G x is a subgroup. Thus the fixed-points are the points x with G x = G. The action is trivial if every point is a fixed point. At the opposite extreme, the action is free if G x = {e} for all x X. Define an equivalence relation on X by x y if there exists g G such that gx = y. A equivalence class is called an orbit, usually denoted O. The orbit determined by a particular x X is denoted O x 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 gx factors through a G-bijection G/G = x O = Gx. Hence [G : G x ] = 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 = O O = x G\X [G : 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 : G x ] = O is independent of the choice. 2 Examples 1. The symmetric group S n acts on the left of [n] := {1, 2,..., n} by permutions. The action is transitive, with the isotropy group of any point isomorphic to S n 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 space H\G of the left action is by definition the set of right cosets 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 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 1 xg. The fixed-point set of either action is the center 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 C G x, 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 conjugacy class of x, and Conj G to mean the set of conjugacy classes. We then have G = x Conj G κ(x) = x Conj G [G : C G x]. 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 subgroups 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 N G H 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 1 x). 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 1 x). The fixed-point set of the combined left action gφg 1 is the subset of G-equivariant maps X Y, as is easily checked. 6. Let X be a set, X n the n-fold Cartesian product of X with itself. Then S n acts on X n by permuting the coordinates. This is a right action, given explicitly by (x 1,..., x n ) σ = (x σ(1),...x σ(n) ). One could check directly that this is a right action, but easier is to note that X n = F ([n], X), where S n 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 topology and geometry 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 topological space, 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 vector space over F. (In fact the finite-dimensionality isn t necessary, but I prefer to avoid distractions.) The projective space P(V ) is the set of lines through the origin in V. Even though at the moment we are not giving 3

4 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 scalar multiplication. This is a very handy interpretation. Second, GL(V ) acts transitively on it: By elementary linear algebra, any line 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 homogeneous space ). 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 L 0. The isotropy group H consists of invertible transformations preserving L 0, i.e. the set of A GL(V ) such that A has an eigenvector in L 0. Then there is a G-isomorphism GL(V )/H = P(V ). If we want to be more explicit, we choose a basis e 1,..., e n and take L 0 = e 1. Then H corresponds to the group of matrices with a i1 = 0 for i > 1 (i.e. the first column is zero except for a 11 ). 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 ). Representation theory 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. precisely the situation one studies in Galois theory. This is 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 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 topological group 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 product topology). This is usually a much stronger assumption than merely saying that G is acting by homeomorphisms. 4

5 There are many variants on this theme: groups acting on metric spaces by isometries, smooth Lie groups actions on smooth manifolds (studied in our Manifolds course), simplicial 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 group homomorphism. 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 S n 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 G x 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 normal subgroup 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 = S n, so we ll think of ρ as a homomorphism G S n. 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 S n. Thus every finite group is isomorphic to a subgroup of some S n, a result known as Cayley s theorem. The n obtained, however, is n = G, and S n 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 translation 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 GL 2 F 2. It acts linearly on F 2 2, 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 homomorphism ρ : GL 2 F 2 S 3. 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 automorphism group of this structure. In example 1 we get ρ : G Aut grp X, 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 Aut C X. 5

6 2. Suppose X 1, X 2 are G-sets with the same underlying set X but with different G-actions. Let ρ 1, ρ 2 be the corresponding homomorphisms G P erm X. Then X 1 is isomorphic to X 2 as a G-set if and only if ρ 1, ρ 2 are conjugate as homomorphisms to P erm X. (Exercise. By conjugate I mean there is a σ P erm X such that σρ 1 σ 1 = ρ 2. ) 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 kernel 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 general linear group, denoted P GL n F. 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/G = x 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. Proof: Suppose x, y X. Choose g G with gx = y. Then gg x g 1 = G y, as is readily checked. Hence any two isotropy groups are conjugate. Conversely, let H be a subgroup conjugate to G x ; say gg x g 1 = H. Then H = G gx, 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 1 Kx 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 := B n F. Then the (B, B)-double cosets of GL n F are given by where W = S n is the Weyl group. GL n F = w W 6 BwB,

7 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 GL n F/B is a topological space known as a flag manifold 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 = S P mod p. Proof: S = 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}. fixed-point, so G has non-trivial center. By the corollary it has a The proposition in turn has a nice corollary: Corollary 6.4 If G has order p 2 (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 quotient group, G is abelian (why?). The next application can be proved using the binomial theorem, but we can also deduce it from Theorem 6.1. Proposition 6.5 Let n = sp k. Then ( n p k ) = s mod p. 7

8 Proof: Partition [n] into s disjoint subsets A 1,..., A s of size p k. Let C i S n be a cyclic group of order p k that permutes A i transitively and fixes the other A j s pointwise, and take P = C 1... C s. Then the action of P on P p k[n] has exactly s fixed points, namely A 1,..., A s. Hence the proposition follows from the theorem. Let G be a finite group, and write G = sp k with s prime to p. A p-sylow subgroup is a subgroup P of order p k. 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 G x 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 G x are p-groups. Then by (i) there is an orbit O with O prime to p. Choose x O; then G x = p i for some i by (ii). But G = G x O, forcing i = k. Hence G x is a p-sylow subgroup. It remains to exhibit such an S. Take S to be the set of subsets of G of size p k, with action induced by the left translation action of G on itself. Then (i) holds by Proposition 6.5. Now let A S. Then G A acts freely on the elements of A (since the left translation action is free), so G A divides p k, proving (ii). The second Sylow theorem says that all p-sylow subgroups are conjugate. precisely: More 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 1 Hx 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 N G P. Hence the number of p-sylow subgroups is [G : N G P ]. This brings us to the third Sylow theorem: Theorem 6.8 Let n p G denote the number of distinct p-sylow subgroups of G. Then n p G divides G and n p G = 1 mod p. Proof: Since n p G = [G : N G P ] for any choice of p-sylow subgroup P, n p G 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 N G P (since it is a normal p-sylow subgroup of N G P ). Now consider the left translation action of P on G/N G P. If P xn G P = xn G P then x 1 P x N G P, forcing x 1 P x = P since P is the unique p-sylow subgroup of N G P. Hence x N G P ; in other words, the P -action has a unique fixed point. Using Theorem 6.1 we conclude 8

9 n p G = G/N G P = (G/N G P ) P = 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 N G P H. Then N G H = H. (In particular, this is true for H = N G P.) Proof: Suppose ghg 1 = H. Then gp g 1 is a p-sylow subgroup of H, so by the second Sylow theorem there is an h H such that hp h 1 = gp g 1. Then h 1 g N G P, 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 X H for X regarded as an H-set, and call X X H the restriction functor. Disjoint unions. Suppose X and Y are G-sets. The disjoint union X Y is a G-set in the obvious way, and in fact is the categorical coproduct. Similarly if X α is any collection of G- sets indexed by a set J, α J X α is a G-set and is the categorical coproduct. The fixed-point set and orbit set functors 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 = X G 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 g 1 [g, x] = [g 1 g, 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 G H X defines the induction functor H-set G-set (the definition of the functor on morphisms 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 category theory notes for a general discussion of such properties). We keep the above notation. 9

10 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!ψ Y φ 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 Hom G (G H X, Y ) = Hom H (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 natural transformation, 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- invariant 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 x 1, x 2 X and gx 1 = x 2, 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. Example. Let F be a field, G = GL n F, and Y the set of pairs (L, v) with L a line in F n and v L. We have an evident G-action on Y given by g (L, v) = (gl, gv). Let L 0 denote the line spanned by the standard basis vector e 1, and let X = {(L, v) Y : L = L 0 }. Then X is invariant under H := {g GL n F : gl 0 = L 0 }. Now let s check conditions (i) and (ii) of the recognition principle above: (i) is clear, since by linear algebra GL n F acts transitively 10

11 on the lines. And if (gl 0, gv) = (L 0, 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 L 0, it is the linear action that pulls back scalar multiplication along the homomorphism H F taking a matrix A to a 11, the upper left entry: A (L 0, v) = (L 0, a 11 v). 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: Proposition 8.1 Suppose H 1,..., H n are normal subgroups of G such that (i) H i ( j i H j ) = {e} and (ii) G = H 1...H n. Then the multiplication map m : H 1 H 2... H n G is an isomorphism of groups. Proof: First note that m is a group homomorphism: For this, one needs to know that for i j the elements of H i, H j commute with one another. But if x H i and y H j, then by normality xyx 1 y 1 H i H j = {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 semidirect product, 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 ρ K: As a set it is just H K, but with group multiplication defined by (h 1, k 1 ) (h 2, k 2 ) = (h 1 (ρ(k 1 )(h 2 )), k 1 k 2 ). The proof that this multiplication defines a group structure is left as an exercise. Note that (h 1, e) (e, k 2 ) = (h 1, k 2 ). 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

12 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 ρ K, and that H ρ 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 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 C 7 is cyclic of order 6, so we can choose an injective homomorphism ρ : C 3 Aut C 7 (there are two such homomorphisms) and form C 7 ρ C 3. 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. Then G = H ρ K, where ρ(k)(h) = khk 1. 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 C n C 2, where C 2 acts on C n as multiplication by S n = A n C 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 T v : x x + v. The group of translations is isomorphic to the additive group of V, and Aff(V ) = V GL(V ). Here the action of GL(V ) on V by conjugation is the same as the standard action. When V = F n, we also write Aff n F in place of Aff(F n ). 4. Consider the symmetric group S p and the p-sylow subgroup C p generated by the standard p-cycle (12...p). Then the normalizer N Sp C p is isomorphic to Aff(F p ) and hence is a semi-direct product of the form C p C p B n F = U n F D n F. What is the action of D n F on U n F? 6. N n F = D n F S n, where S n acts on D n F = (F ) n by permuting the factors. This is a type of semi-direct product known as a wreath product, disussed further in the exercises. 12

13 8.3 Group extensions A group extension 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 C 2 G C 2, then since G = 4 we know G is abelian, but without further information there is no way to know whether G is C 4 or C 2 C 2. The extension is central if H C(G). Note, for example, that if H = C 2 then the extension is automatically central. This leads to another example of the ambiguity inherent in group extensions: If we have an extension C 2 G C 2 C 2, then it is a central extension but G could be any of the five groups of order 8 except C 8 : (C 2 ) 3, C 2 C 4, the dihedral group D 8, or the quaternion group Q 8. Each of these four groups fits into such an extension, as you can easily check. Note that a semi-direct product G := H K fits into an extension H G K. In fact we can characterize the semi-direct products in terms of extensions. A group extension H i G π K splits if there is a homomorphism s : K G such that π s = Id K. We call s a splitting (or sometimes a section ) of π. Proposition 8.3 If G := H K is a semi-direct product, the extension H G K splits. Conversely if H G K is a split group extension, then G = H K 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 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 K G such that π : G K maps K 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 C q C p. 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 C q G π C p. Now choose a p-sylow subgroup H. Then π H is injective, and hence an isomorphism. This proves the claim. Note that C p acts on C q by some homomorphism C p Aut C q = 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 equivalently, finding a normal subgroup H with quotient K one simple possibility we might hope 13

14 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 S 4 in two steps out of abelian groups. First we form the extension C 2 2 A 4 C 3. Then we form the extension A 4 S 4 C 2. But we need a smoother way to think about building 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} G 1 G 2 G 3... A decreasing filtration likewise consists of subgroups G G 1 G 2 G 3... In either case the filtration stabilizes at G n if G k = G n for all k n. An increasing (resp. decreasing) filtration is finite if G n = G for some n (resp. G n = {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 abelian group A has a natural decreasing p-adic filtration A pa p 2 A... (which need not terminate; think of A = Z), as well as a natural p-torsion filtration A[p] A[p 2 ]... (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 G k /G k 1 (increasing case) or G k /G k+1 (decreasing case) have some simple form that we understand; then the hope is that we can recover, perhaps by an 14

15 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 G i is normal in G i+1 for all i, and normal if G i 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 A 4 ). 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 increasing) of H and K. Then we get a filtration of G by splicing them together: {e} = H 0 H 1... H π 1 K 1...π 1 K n = 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, A 4 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 G 0 = G and G i+1 = [G i, G i ]. We call this the commutator 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 simple group 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

16 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 = H 0 H 1... H m = {e} be a solvable filtration. Then G/H 1 is abelian, so [G, G] H 1. Similarly, since H 1 /H 2 is abelian, we have G (2) [H 1, H 1 ] H 2. Continuing in this manner, we find that G (k) H k 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 G 1 G 2... be a solvable filtration for G, and consider the filtration π(g i ) of H. It is a finite subnormal filtration, and the quotients are abelian because π(g i )/π(g i+1 ) is a quotient of G i /G i+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 ring, but to avoid distractions we will stick to the case of a field F. Recall that B n F GL n F is the Borel subgroup of upper triangular matrices. Proposition 9.3 B n F is solvable. Proof: Note that B 1 F = F, and that there is a surjective homomorphism π : B n F B n 1 F given by simply projecting A B n F onto its upper left (n 1) (n 1) block. By induction we can assume B n 1 F 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 columns, which we will quaintly denote RC n F. It fits into an extension RCn RC u n F, where the second map is just projection on the nn coordinate and RCn u is therefore the 16

17 subgroup with a nn = 1 (the superscript u if for unipotent ). One easily checks that RC u n is isomorphic to the additive group F n 1, and in particular is abelian. So RC n F is solvable and the proof is complete. See the exercises for a discussion of the commutator series of B n F. 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 p a q b theorem. Theorem 9.5 Suppose p, q are primes and G = p a q b. 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 = is the smallest number that is neither the product of three distinct primes nor of the form p a q b. (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 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} = G 0 G 1... of G is central if G i+1 /G i C(G/G i ). 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, S 3 is solvable but not nilpotent. Proposition 9.8 Every finite p-group is nilpotent. 17

18 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 ascending central series, defined recursively as follows: Let C 1 = C(G). Having defined the normal subgroup C k, let π : G G/C k be the quotient homomorphism, and set C k+1 = π 1 C(G/C k ). It is clear that C k+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 G i with G m = G and show inductively that G i C i. Then C m = 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} G 1... G m = G such that (i) each G i is invariant under the K-action, and (ii) the induced K-action on each quotient G i /G i 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 matrices U n F. For G we take the additive group F n, with its standard left U n F 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 U n F. 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 S 3, 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 C 3 S 3 C 2 is not nilpotent. Proposition 9.10 The class of nilpotent groups is closed under subgroups, quotients, nilpotent 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

19 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 nilpotently 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 U n F is the group of upper triangular n n matrices with 1 s on the diagonal. Since it is a subgroup of the solvable group B n F, it is solvable. But more is true: Proposition 9.11 U n F is nilpotent. Proof: We follow closely the proof already given for solvability of B n F. As in that case, there is a group extension RC u nf U n F U n 1 F, where the second map is projection on the upper left (n 1) (n 1) block and RC u nf is again the unipotent right column group ; for example RC 3 F consists of matrices 1 0 a 0 1 b By induction we can assume U n 1 F is solvable, so it suffices to show that the action of U n F on RC u nf by conjugation is nilpotent. Since RC u nf is abelian, the action of U n F factors through U n 1 F, so what we need to show is that the conjugation action of U n 1 F on RC u nf is nilpotent. But under the evident isomorphism RC u nf = F n 1 (where the entries of the right column are ordered from top to bottom), this action corresponds to the standard linear action of U n 1 F on F n 1. 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) {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 H 1 on which G acts trivially, so H 1 H C(G). 19