Let $G$ be a finite $T$-group (i.e. a group in which every subnormal subgroup is normal). Assume that $L$ is a normal abelian subgroup of odd order of $G$ and suppose that $G/L$ is a Dedekind group. Let now $L_p$ and $L_{p'}$ denote the Sylow $p-$ and Sylow $p'-$subgroups of L. Let $M(p)/L$ be the Sylow $p-$subgroup of $G/L$. Why is $M(p)/L_p'$ a Dedekind p-group?
This is a part of a proof of a theorem. It seems that $L$ has only a $p$-Sylow and a $p'$-Sylow, i.e. $|L|=p^k\cdot(p')^j$ for some positive natural $k,j$. Is it true? Why?