Abstract Algebra: Theory and Applications · Problem 11.4.17

Matrix Groups and Symmetry — Thomas Judson

Mathematics — Abstract Algebra · surjective-homomorphism-quotient-groups

Problem

Let

\phi : G_1 \to G_2

be a surjective group homomorphism. Let

H_1

be a normal subgroup of

G_1

and suppose that

\phi(H_1) = H_2

. Prove or disprove that

G_1/H_1 \cong G_2/H_2

. Hint. Find a counterexample.