Abstract Algebra: Theory and Applications · Problem 15.4.23

Rings — Thomas Judson

Mathematics — Abstract Algebra · sylow-theorems

Problem

Prove that the number of distinct conjugates of a subgroup

H

of a finite group

G

[G : N(H)]

. Hint. Define a mapping between the right cosets of

N(H)

G

and the conjugates of

H

G

N(H)g \mapsto g^{-1}Hg

. Prove that this map is a bijection.