Abstract Algebra: Theory and Applications · Problem 11.5.5

Matrix Groups and Symmetry — Thomas Judson

Mathematics — Abstract Algebra · inner-automorphism-homomorphism-center

Problem

Let

G

be a group and

i_g

be an inner automorphism of

G

, and define a map

G \to \text{Aut}(G)

g \mapsto i_g.

Prove that this map is a homomorphism with image

\text{Inn}(G)

and kernel

Z(G)

. Use this result to conclude that

G/Z(G) \cong \text{Inn}(G).