Abstract Algebra: Theory and Applications · Problem 13.18

Group Actions — Thomas Judson

Mathematics — Abstract Algebra · solvable-groups

Problem

Prove that

G

is a solvable group if and only if

G

has a series of subgroups

G = P_n \supset P_{n-1} \supset \cdots \supset P_1 \supset P_0 = \{e\}

where

P_i

is normal in

P_{i+1}

and the order of

P_{i+1}/P_i

is prime.