I'm sorry, I think I left some ambiguity as to what assumption I can make. According to my text (Artin), a Sylow

-subgroup of a group

, where

is a subgroup of

that has order

.

That said, the way the first Sylow theorem is stated in my book (and the way my class is using it) is as follows:

A finite group whose order is divisible by a prime

contains a Sylow

-subgroup.

So I feel like what you are suggesting depends on my

*knowing* that there is a subgroup of order

where

. Based on the way we (my class/professor) are defining the first Sylow theorem, I don't think I can assume this. I realize that some books state the first Sylow theorem differently, and if I could apply that statement, then this problem would be proven just as you have suggested.