# Thread: Showing G is Abelian

1. ## Showing G is Abelian

Let $\displaystyle p$ & $\displaystyle q$ be primes with $\displaystyle q<p$ and suppose $\displaystyle G$ is a group of order$\displaystyle p^2q$. Suppose that $\displaystyle G$ has unique subgroup of order $\displaystyle q$. Prove that $\displaystyle G$ is abelian stating any theorems you use...

I'm thinking I have to use some form of Sylows theorem to show that $\displaystyle G$ is a direct sum of p-groups and hence Abelian but no idea how (or even if that is the tactic)!!

Any help appreciated...

2. ## Re: Showing G is Abelian

Work with the class equation. It involves some work, but it's not dificult.

3. ## Re: Showing G is Abelian

Oh yeah, before that you need to apply the 3rd Sylow theorem to show that there's only 1 p-Sylow subgroup.

4. ## Re: Showing G is Abelian

Hmmm, I've had a think and come up with this but it isn't really using the class equation I don't think!

Using Sylow III I can certainly see that $\displaystyle n_p = 1$ as $\displaystyle n_p|q$ and $\displaystyle n_p=1(modp)$ and as $\displaystyle q<p, n_p =1$

Also from Sylow II we know that as $\displaystyle n_p = 1$ then$\displaystyle P \triangleleft G$ and $\displaystyle P$ has order $\displaystyle p^2$ thus Abelian. The intersection or $\displaystyle P$ and $\displaystyle Q$ is $\displaystyle {e}$ so $\displaystyle G$ is the direct product of the two abelian groups hence G is Abelian

5. ## Re: Showing G is Abelian

Yes, that works too, as long as you show that G=<QUP>, which, with what you know, is imediate.