# Thread: Prove G is a solvable group

1. ## Prove G is a solvable group

Hi there,

I am trying to solve this problem:

Suppose G is a finite group, and the order of G is ${p_1}^2 {p_2}^2$, where $p_1,p_2$ are primes such that ${p_1}^2 < p_2$. Show that G is a solvable group.

I've tried various approaches through Sylow groups and normal subgroups etc but haven't been able to do it, can anyone lend a hand?

Many thanks!

2. Let's look at the number of Sylow $p_2$-subgroups. By the Sylow theorems, since $n_{p_2}=kp_2+1$ and $n_{p_2}|p_1^2$, $n_{p_2}$ must be 1. Thus, the Sylow $p_2$-subgroup, $P$ is normal in $G$. Now consider

$1 \unlhd P \unlhd G$.

Any group of order $p^2$ (for p prime) is abelian, and since $|G/P|=p_1^2,$ $G/P$ is abelian.