Abstract Algebra: Theory and Applications · Problem 15.4.8

Rings — Thomas Judson

Mathematics — Abstract Algebra · sylow-theorems

Problem

Let

G

be a group of order

p^2 q^2

, where

p

and

q

are distinct primes such that

q \nmid p^2 - 1

and

p \nmid q^2 - 1

. Prove that

G

must be abelian. Find a pair of primes for which this is true. Hint. Show that

G

has a normal Sylow

p

-subgroup of order

p^2

and a normal Sylow

q

-subgroup of order

q^2