Abstract Algebra: Theory and Applications · Problem 15.4.21

Rings — Thomas Judson

Mathematics — Abstract Algebra · sylow-theorems

Problem

The Frattini Lemma. If

H

is a normal subgroup of a finite group

G

and

P

is a Sylow

p

-subgroup of

H

, for each

g \in G

show that there is an

h

H

such that

gPg^{-1} = hPh^{-1}

. Also, show that if

N

is the normalizer of

P

, then

G = HN