Semidirect Products, Extensions and Cohomological Classification

If K and Q are groups, then an extension of K by Q is a short exact sequence 1 →K i→G p→Q →1. The notation K is to remind us of kernel, and the notation Q is to remind us of quotient. There is an alternative usage of the term extension, which calls the (middle) group G (not the short exact sequence) an extension if it contains a normal subgroup K1 with K1 ∼= K and G/K1 ∼= Q. As do most people, we will use the term in both senses.

Recall that an automorphism of a group K is an isomorphism K →K. The automorphism group, denoted by Aut(K), is the group of all the automorphisms of K with composition as operation. Of course, extensions are defined for arbitrary groups K, but we are going to restrict our attention to the special case when K is abelian. If G is an extension of K by Q, it would be confusing to write G multiplicatively and its subgroup K additively. Hence, we shall use the following notational convention: Even though G may not be abelian, additive notation will be used for the operation in G. Corollary 10.4 gives the main reason for this decision.

Let K be a Q-module. An extension G of K by Q realizes the operators if, for all x ∈Q and a ∈K, we have xa = ℓ(x) + a −ℓ(x); that is, the given scalar multiplication of ZQ on K coincides with the scalar multiplication of Corollary 10.4 arising from conjugation. Here is the construction.

Given a lifting ℓ: Q →G, with ℓ(1) = 0, of an extension G of K by Q, then a factor set5 (or cocycle) is a function f : Q × Q →K such that ℓ(x) + ℓ(y) = f (x, y) + ℓ(xy) for all x, y ∈Q. It is natural to choose liftings with ℓ(1) = 0, and so we have incorporated this condition into the definition of factor set; our factor sets are often called normalized factor sets. Of course, a factor set depends on the choice of lifting ℓ. When G is a split extension, then there exists a lifting that is a homomorphism; the corresponding factor set is identically 0. Therefore, we can regard a factor set as the obstruction to a lifting being a homomorphism; that is, factor sets describe how an extension differs from being a split extension.

Given a group Q and a Q-module K, a function g : Q × Q →K is called a coboundary if there exists a function h : Q →K with h(1) = 0 such that, for all x, y ∈Q, g(x, y) = xh(y) −h(xy) + h(x). The term coboundary arises because its formula is an alternating sum analogous to the formula for geometric boundaries that we described in Section 10.1. We have just shown that if f and f ′ are factor sets of an extension G that arise from different liftings, then f ′ −f is a coboundary.

Given a group Q and a Q-module K, two extensions G and G′ of K by Q that realize the operators are called equivalent if there is a factor set f of G and a factor set f ′ of G′ so that f ′ −f is a coboundary.

If Q is a group and K is a Q-module, define H1(Q, K) = Der(Q, K)/PDer(Q, K), where PDer(Q, K) is the subgroup of Der(Q, K) consisting of all the principal derivations.

A projective resolution of a module M is an exact sequence, · · · →Pn →Pn−1 →· · · →P1 →P0 →M →0, in which each module Pn is projective. A free resolution is a projective resolution in which each module Pn is free.

Let 0 →K i→G p→Q →1 be an extension of an abelian group K by a group Q, and let ℓ: Q →G be a lifting. (i) For every x ∈Q, conjugation θx : K →K, defined by θx : a ↦ℓ(x) + a −ℓ(x), is independent of the choice of lifting ℓ(x) of x. [For convenience, we have assumed that i is an inclusion; this merely allows us to write a instead of i(a).] (ii) The function θ : Q →Aut(K), defined by x ↦θx, is a homomorphism.

If 0 →K i→G p→Q →1 is an extension of an abelian group K by a group Q, then K is a left ZQ-module if we define xa = ℓ(x) + a −ℓ(x), where ℓ: Q →G is a lifting, x ∈Q, and a ∈K; moreover, the scalar multiplication is independent of the choice of lifting ℓ.

Given a group Q and a Q-module K, then G = K ⋊Q is a semidirect product of K by Q that realizes the operators.

Let Q be a group, K a Q-module, and 0 →K →G →Q →1 an extension realizing the operators. (ii) the cocycle identity holds: For all x, y, z ∈Q, we have f (x, y) + f (xy, z) = x f (y, z) + f (x, yz).

Let Q be a group, let K be a Q-module, and let G be an extension of K by Q realizing the operators. Then there exists a factor set f : Q × Q →K with G ∼= G(K, Q, f ).

Given a group Q and a Q-module K, then Z2(Q, K) is an abelian group with operation pointwise addition, f + f ′ : (x, y) ↦f (x, y) + f ′(x, y), and B2(Q, K) is a subgroup of Z2(Q, K).

Let Q be a group, let K be a Q-module, and let e(Q, K) denote the family of all the equivalence classes of extensions of K by Q realizing the operators. There is a bijection ϕ : H2(Q, K) →e(Q, K) that takes 0 to the class of the split extension.

First, ϕ is a well-defined injection: f and g are factor sets with f + B2 = g + B2 if and only if [G(K, Q, f )] = [G(K, Q, g)], by Proposition 10.17. By Theorem 10.14 and the remark following it, [G] = [G(K, Q, f )] for some factor set f , and so [G] = ϕ( f + B2). Finally, the zero factor set corresponds to the semidirect product. • If H is a group and if there is a bijection ϕ : H →X, where X is a set, then there is a unique operation defined on X making X a group and ϕ an isomorphism: Given x, y ∈X, there are g, h ∈H with x = ϕ(g) and y = ϕ(h), and we define xy = ϕ(gh).

Let G be a finite group of order mn, where (m, n) = 1. If K is an abelian normal subgroup of order m, then K has a complement and G is a semidirect product.

If a finite group G has a normal maximal p-subgroups P, for some prime divisor p of |G|, then G is a semidirect product; more precisely, P has a complement.

If Q is a group, K is a Q-module, and 0 →K →G →Q →1 is a split extension, then there is an isomorphism Stab(Q, K) →Der(Q, K).

Let 0 →K →G →Q →1 be a split extension, and let C and C′ be complements of K in G. If H1(Q, K) = {0}, then C and C′ are conjugate.

Let G be a finite group of order mn, where (m, n) = 1. If K is an abelian normal subgroup of order m, then G is a semidirect product of K by G/K, and any two complements of K are conjugate.

For any group Q, there is an isomorphism ZQ/ im d1 ∼= Z, where Z is regarded as a trivial Q-module.

(i) The direct product K × Q is an extension of K by Q; it is also an extension of Q by K. (ii) Both S3 and I6 are extensions of I3 by I2. On the other hand, I6 is an extension of I2 by I3, but S3 is not, for S3 contains no normal subgroup of order 2. ◀ We have just seen, for any given ordered pair of groups, that there always exists an extension of one by the other (their direct product), but there may be other extensions as well. The extension problem is to classify all possible extensions of a given pair of groups K and Q. If a group G has a composition series G = K0 ≥K1 ≥K2 ≥· · · ≥Kn−1 ≥Kn = {1} with simple factor groups Q1, . . . , Qn, where Qi = Ki−1/Ki for all i ≥1, then G could be recaptured from Qn, Qn−1, . . . , Q1 by solving the extension problem n times. Now all finite simple groups have been classified, and so we could survey all finite groups if we could solve the extension problem. Let us begin by recalling the partition of a group into the cosets of a subgroup. We have already defined a transversal of a subgroup K of a group G as a subset T of G consisting of exactly one element from each coset3 Kt of K.

(i) If Q is a group and K is a Q-module, then a function u : Q →K of the form u(x) = xa0 −a0, where a0 ∈K, is a derivation: u(x) + xu(y) = xa0 −a0 + x(ya0 −a0) = xa0 −a0 + xya0 −xa0 = xya0 −a0 = u(xy). 8Earlier, we defined a derivation of a (not necessarily associative) ring R as a function d : R →R with d(xy) = d(x)y + xd(y). Derivations here are defined on modules, not on rings. General Extensions and Cohomology A derivation u of the form u(x) = xa0 −a0 is called a principal derivation. If the action of Q on K is conjugation, xa = x + a −x, then xa0 −a0 = x + a0 −x −a0; that is, xa0 −a0 is the commutator of x and a0. (ii) It is easy to check that the set PDer(Q, K) of all the principal derivations is a subgroup of Der(Q, K). ◀ Recall that Stab(Q, K) denotes the group of all the stabilizing automorphisms of an extension of K by Q.