STRUCTURE OF GROUPS

Excerpted from Abstract Algebra: Supplementary Lecture Notes, copyright 1999 by John Beachy.
7.8 Nilpotent groups
7.9 Semidirect products
7.10 Classification of groups of small order
Some group multiplication tables

Nilpotent groups

We now define and study a class of solvable groups that includes all finite abelian groups and all finite p-groups.

7.8.1 Definition. For a group G we define the ascending central series
Z1(G) Z2(G) . . . of G as follows:
Z1(G) is the center Z(G) of G;
Z2(G) is the unique subgroup of G with
Z1(G) Z2(G) and Z2(G) / Z1(G) = Z(G/Z1(G)).
We define Zi(G) inductively, in the same way.
The group G is called nilpotent if there exists a positive integer n with Zn(G)=G.

7.8.2 Proposition. If G1, G2, . . . , Gn are nilpotent groups, then so is
G = G1 × G2 × . . . × Gn.

7.8.3 Lemma. If P is a Sylow p-subgroup of a finite group G, then the normalizer N(P) is equal to its own normalizer in G.

7.8.4 Theorem. The following conditions are equivalent for any finite group G.
(1) G is nilpotent;
(2) no proper subgroup H of G is equal to its normalizer N(H);
(3) every Sylow subgroup of G is normal;
(4) G is a direct product of its Sylow subgroups.

7.8.5 Corollary. Let G be a finite nilpotent group of order n. If m is any divisor of n, then G has a subgroup of order m.

7.8.6 Lemma. [Frattini's Argument] Let G be a finite group, and let H be a normal subgroup of G. If P is any Sylow subgroup of H, then G=HN(P), and [G:H] is a divisor of |N(P)|.

7.8.7 Proposition. A finite group is nilpotent if and only if every maximal subgroup is normal.

Semidirect products

7.9.1 Definition. Let G be a group with subgroups N and K such that
(i) N is normal in G;
(ii) N K = {e}; and
(iii) NK = G.
Then G is called the semidirect product of N by K.

Example. 7.9.6. (Dn is a semidirect product.)
Consider the dihedral group Dn, described by generators a of order n and b of order 2, with the relation ba=a-1b. Then <a> is a normal subgroup, <a><b>={e}, and <a><b>=Dn. Thus the dihedral group is a semidirect product of a cyclic subgroups of order n by a cyclic subgroup of order 2.

Definition 7.9.1 describes an ``internal'' semidirect product. We now use the automorphism group to give a general definition of an ``external'' semidirect product.

7.9.2 Definition. Let G be a multiplicative group, and let X be an abelian group, denoted additively. Let :G->Aut(X) be a group homomorphism. The semidirect product of X and G relative to is
X G = { (x,a) | x X, a G }
with the operation (x1, a1) (x2, a2) = ( x1 + (a1) [x2], a1 a2), for x1, x2 X and a1, a2 G.

7.9.3 Proposition. Let G be a multiplicative group, let X be an additive group, and let :G->Aut(X) be a group homomorphism.
(a) The semidirect product X G is a group.
(b) The set { (x,a) X G | x = 0 } is a subgroup of X G that is isomorphic to G.
(c) The set N = { (x,a) X G | a = e } is a normal subgroup of X G that is isomorphic to X, and (X G)/N is isomorphic to G.

We can now give a more general definition. We say that Zn Znx is the holomorph of Zn, denoted by Hn.

7.9.4 Definition. Let G be a group and let X be an abelian group. If G acts on X and a(x+y)=ax+ay, for all aG and x,yX, then we say that G acts linearly on X.

7.9.5 Proposition. Let G be a group and let X be an abelian group. Then any group homomorphism from G into the group Aut(X) of all automorphisms of X defines a linear action of G on X. Conversely, every linear action of G on X arises in this way.

7.9.6 Proposition. Let G be a multiplicative group with a normal subgroup N, and assume that N is abelian. Let :G->G/N be the natural projection. The following conditions are equivalent:
(1) There exists a subgroup K of G such that NK={e} and NK=G;
(2) There exists a homomorphism :G/N->G such that is the identity on G/N;
(3) There exists a homomorphism :G/N->Aut(N) such that N (G/N)G.

Classification of groups of small order

In this section we study finite groups of a manageable size. Our first goal is to classify all groups of order less than 16 (at which point the classification becomes more difficult). Of course, any group of prime order is cyclic, and simple abelian. A group of order 4 is either cyclic, or else each nontrivial element has order 2, which characterizes the Klein four-group.

7.10.1 Proposition. Any nonabelian group of order 6 is isomorphic to S3.

7.10.2 Proposition. Any nonabelian group of order 8 is isomorphic either to D4 or to the quaternion group Q.

7.10.3 Proposition. Let G be a finite group.
(a) Let N be a normal subgroup of G. If there exists a subgroup H such that HN={e} and |H|=[G:N], then GN H.
(b) Let G be a group with |G|=pnqm, for primes p,q. If G has a unique Sylow p-subgroup P, and Q is any Sylow q-subgroup of G, then GP Q. Furthermore, if Q' is any other Sylow q-subgroup, then P Q' is isomorphic to P Q.
(c) Let G be a group with |G|=p2q, for primes p,q. Then G is isomorphic to a semidirect product of its Sylow subgroups.

7.10.4 Lemma. Let G,X be groups, let , :G->Aut(X), and let , be the corresponding linear actions of G on X.
Then X G X G if there exists Aut(G) such that = .

7.10.5 Proposition. Any nonabelian group of order 12 is isomorphic to one of
A4, D6, or Z3 Z4.

The following table summarizes the above information on the isomorphism classes of groups of order less than sixteen.
Order 2: Z2
Order 3: Z3
Order 4: Z4, Z2 × Z2
Order 5: Z5
Order 6: Z6, S3
Order 7: Z7
Order 8: Z8, Z4 × Z2, Z2 × Z2 × Z2, D4, Q
Order 9: Z9, Z3 × Z3
Order 10: Z10, D5
Order 11: Z11
Order 12: Z12, Z6 × Z2 A4, D6, Z3 Z4
Order 13: Z13
Order 14: Z14, D7
Order 15: Z15

7.10.6 Proposition. Let G be a finite simple group of order n, and let H be any proper, nontrivial subgroup of G.
(a) If k = [G:H], then n is a divisor of k!.
(b) If H has m conjugates, then n is a divisor of m!.

7.10.7 Proposition. The alternating group A5 is the smallest nonabelian simple group.