If all Sylow subgroups are normal then the group is solvable

Question

Show: all $p$-Sylow groups are normal subgroups $\implies$ group $G$ is solvable.

I know that all subgroups of the different $p$-Sylow groups are solvable, but do not know if this helps.

Other idea is to show that all $p$-Sylow groups are solvable and that the factor groups are solvable as well, then it follows, that $G$ is solvable. But I don't know why $p$-Sylow groups should be solvable.

Answer 1

Try to prove by induction the following easy

Claim: is $\;G\;$ is a finite $\,p$-group, say $\,|G|=p^n\;$ , then

$$\forall 0\le k\le n\;\;\exists\,H_k\lhd G\;\;s.t.\,\,|H_k|=p^k$$

With the above you're done since then $\,1\lhd H_1\lhd\ldots\lhd H_n=G\;$ is an abelian series for $\,G\,$ ...

Answer 2

Here's a hint:

If two subgroups $H$ and $K$ are normal and intersect precisely in $\{e\}$, then $HK = KH = H \times K$. What is the intersection of a Sylow $p$-subgroup and a Sylow $q$-subgroup for different $p$ and $q$?