232 CHAPTER 9. SOME ITEMS WHICH RESEMBLE LINEAR ALGEBRA

Definition 9.4.40 A group G is solvable if there exists a decreasing sequence of subgroups{Hi}m

i=0 such that Hi is a normal subgroup of H(i−1),

G = H0 ⊇ H1 ⊇ ·· · ⊇ Hm = {ι} ,

and each quotient group Hi−1/Hi is Abelian. That is, for [a] , [b] ∈ Hi−1/Hi,

[ab] = [a] [b] = [b] [a] = [ba]

Note that if G is an Abelian group, then it is automatically solvable. In fact you can justconsider H0 = G,H1 = {ι}. In this case H0/H1 is just the group G which is Abelian. Also,the definition requires Hm−1 to be Abelian.

There is another idea which helps in understanding whether a group is solvable. Itinvolves the commutator subgroup. This is a very good idea.

Definition 9.4.41 Let a,b ∈ G a group. Then the commutator is

aba−1b−1

The commutator subgroup, denoted by G′, is the smallest subgroup which contains all thecommutators.

The nice thing about the commutator subgroup is that it is a normal subgroup. Thereare also many other amazing properties.

Theorem 9.4.42 Let G be a group and let G′ be the commutator subgroup. Then G′ is anormal subgroup. Also the quotient group G/G′ is Abelian. If H is any normal subgroupof G such that G/H is Abelian, then H ⊇ G′. If G′ = {ι} , then G must be Abelian.

Proof: The elements of G′ are just finite products of things like aba−1b−1. Note thatthe inverse of something like this is also one of these.(

aba−1b−1)−1= bab−1a−1.

Thus the collection of finite products is indeed a subgroup. Now consider h ∈ G. Then

haba−1b−1h−1 = hah−1hbh−1ha−1h−1hb−1h−1

= hah−1hbh−1 (hah−1)−1 (hbh−1)−1

which is another one of those commutators. Thus for c a commutator and h ∈ G,

hch−1 = c1

another commutator. If you have a product of commutators c1c2 · · ·cm, then

hc1c2 · · ·cmh−1 =m

∏i=1

hcih−1 =m

∏i=1

di ∈ G′

where the di are each commutators. Hence G′ is a normal subgroup.Consider now the quotient group. Is [g] [h] = [h] [g]? In other words, is [gh] = [hg]?

In other words, is gh(hg)−1 = ghg−1h−1 ∈ G′? Of course. This is a commutator and Ǵ′

consists of products of these things. Thus the quotient group is Abelian.