Show that a group of order $175$ is not simple.

Question

$|G| = 175 = 5^2 \times 7$
After small calculation I found that only possible value of $n_5 = 1$ and $n_7 = 1$.
How to prove that $G$ is not simple group?

Answer 1

Let $|G| = 175$. You've correctly calculated that $n_5 = 1$ and $n_7 = 1$. Since there is only one Sylow-$25$ subgroup and only one Sylow-$7$ subgroup - both these Sylow subgroups must be normal subgroups of $G$.

Since we have shown the existence of non-trivial normal subgroups, $G$ is not simple.

The following theorem (proof here) will help:

A Sylow $p$-subgroup is unique if and only if it is a normal subgroup.

Answer 2

You proved it is not simple, because you found a nontrivial normal subgroup.

Here's an alternate. By Burnside's theorem the group is solvable. If it's non-abelian it can't be simple.

On the other hand if it's abelian and simple its order would have to be prime.