Abstract Algebra: Theory and Applications · Problem 10.4.11

Homomorphisms — Thomas Judson

Mathematics — Abstract Algebra · normal-subgroups

Problem

If a group

G

has exactly one subgroup

H

of order

k

, prove that

H

is normal in

G

. Hint. For any

g \in G

, show that the map

i_g : G \to G

defined by

i_g : x \mapsto gxg^{-1}

is an isomorphism of

G

with itself. Then consider

i_g(H)