Abstract Algebra: Theory and Applications · Problem 13.25

Group Actions — Thomas Judson

Mathematics — Abstract Algebra · solvable-groups

Problem

Recall that the commutator subgroup

G'

of a group

G

is defined as the subgroup of

G

generated by elements of the form

a^{-1}b^{-1}ab

for

a, b \in G

. We can define a series of subgroups of

G

G^{(0)} = G

G^{(1)} = G'

, and

G^{(i+1)} = (G^{(i)})'

. (a) Prove that

G^{(i+1)}

is normal in

(G^{(i)})'

. The series of subgroups

G^{(0)} = G \supset G^{(1)} \supset G^{(2)} \supset \cdots

is called the derived series of

G

. (b) Show that

G

is solvable if and only if

G^{(n)} = \{e\}

for some integer

n