Abstract Algebra: Theory and Applications · Problem 13.17

Group Actions — Thomas Judson

Mathematics — Abstract Algebra · solvable-groups

Problem

Let

N

be a normal subgroup of

G

. If

N

and

G/N

are solvable groups, show that

G

is also a solvable group. Hint. If

N

and

G/N

are solvable, then they have solvable series

N = N_n \supset N_{n-1} \supset \cdots \supset N_1 \supset N_0 = \{e\}

G/N = G_n/N \supset G_{n-1}/N \supset \cdots G_1/N \supset G_0/N = \{N\}.