Let be a finite group and a prime. Prove that if and , then . Give an example to show that, in general, a Sylow -subgroup of if a subgroup of need not be a Sylow -subgroup of .

If is a Sylow -subgroup of , then does not divide . Now , so that does not divide ; hence is a Sylow -subgroup of .

A trivial counterexample to the converse statement is the subgroup ; is a Sylow 2-subgroup of itself, but clearly not a Sylow 2-subgroup of since .

## Comments

I have another example for you, I’m not 100% this works, though: consider , and . Then can be identified inside as the Sylow 3-subgroup of , however the order of disqualifies it from being the 3-subgroup of .