This question was asked in 
<https://math.stackexchange.com/questions/729648>. Since I did not get any answer I am asking it here.


In an answer in Mathoverflow I see an answer but I could not understand it. May anybody help me for Understanding it. The question and its answer are:


[On the character degrees of a finite group with special structure](https://mathoverflow.net/questions/160427/on-the-character-degrees-of-a-finite-group-with-special-structure)


In the beginning of the answer it is stated that by transfer and a theorem of Gaschutz we see that $N$ is complemented. By Gaschutz lemma (Stellmacher Page 74) we need an abelian and normal subgroup but $N$ is not abelian by assumption. could anybody explain why $N$ has a complement? 
Alex



On the Complement of a subgroup
If a finite group $G$ has an Abelian Sylow $p$-subgroup $P,$ then by transfer, we have $P \cap G^{\prime} \cap Z(G) = 1.$ Your group $N = O\_{p}(N) \times H.$ Since $G/N \cong {\rm PSL}(2,p)$ it now follows that $O\_{p}(N) \leq Z(G).$ Furthermore, $O\_{p}(N)$ is complemented in $G$ by Gaschutz's theorem ($G$ can't have cyclic Sylow $p$-subgroups of order $p^{2},$ again by a transfer argument, since $Z(G)$ contains an element of order $p,$ which lies outside $G^{\prime}).$ The complement $K$ to $O\_{p}(N)$ has $H$ as a normal subgroup, and indeed $H$ is complemented in $K$ by Schur-Zassenhaus (which I should have explicitly mentioned). A complement $C$ to $H$ in $K$ is a complement to $N$ in $G.$


