Abstract Algebra: Theory and Applications · Problem 11.4.14

Matrix Groups and Symmetry — Thomas Judson

Mathematics — Abstract Algebra · canonical-homomorphism-subgroup-order

Problem

Let

G

be a finite group and

N

a normal subgroup of

G

. If

H

is a subgroup of

G/N

, prove that

\phi^{-1}(H)

is a subgroup in

G

of order

|H| \cdot |N|

, where

\phi : G \to G/N

is the canonical homomorphism.