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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

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

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  • $\begingroup$ I think that might be a typo and what he really wanted to write is that $H$ has a complement because $|H|$ and $|G:H|=p|PSL(2,p)|$ are coprime. This follows from the theorem of Schur-Zassenhaus (which can be derived from Gaschütz' theorem I think) $\endgroup$
    – Johannes Hahn
    Commented Mar 28, 2014 at 22:04
  • $\begingroup$ But in the proof it Seems that $N$ need a complement $\endgroup$
    – user48877
    Commented Mar 28, 2014 at 22:18

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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 $G.$

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  • $\begingroup$ l appreciate for Your help. Since I am a beginner in group theory could You let me know what is the transfer argument means and also could You introduce me a reference for it and also for the Gaschutz theorem. $\endgroup$
    – user48877
    Commented Mar 28, 2014 at 22:46
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    $\begingroup$ Most reasonably advanced texts in finite group theory treat transfer (eg Gorenstein, Suzuki). Almost any book on cohomology of finite groups will treat Gaschutz's theorem. $\endgroup$
    – Geoff Robinson
    Commented Mar 28, 2014 at 23:00
  • $\begingroup$ Dear Professor Robinson thanks for Your helps. could you kindly help me the second use of transfer Argument. In fact I can not get this result how using the element of order $ p$ in $ Z ( G)$ which lies outside $ G' $ implies that the Sylow subgroup is not cyclic. $\endgroup$
    – user48877
    Commented Mar 31, 2014 at 19:21

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