A question on solvable groups. Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will we show that.

**Hint** : the Sylow theorems guarantee that a group of order $p^n$ has a chain of subgroups $G_0 \subset \dots \subset G_n$, each normal in the next, with $|G_i|=p^i$. This shows that a group of order $p^n$ is solvable. Now demonstrate such a sequence on $S$ by inducting on $K$, the number of prime power factors.