I want to prove Lemma 2.1(1) in this paper, the first pdf file in the page(adsbygoogle = window.adsbygoogle || []).push({});

This is my proof.

. Since H is X−s−permutable in G, then for P Sylow of G there exists x [itex]\in[/itex] X such that P[itex]^{x}[/itex]H=HP[itex]^{x}[/itex]. The Sylow of N are of the form P∩N. Thus,(P∩N)[itex]^{x}[/itex]H=H(P∩N)[itex]^{x}[/itex]. Hence, H is X−s−permutable in N.

The problem is, according to the definition in the second page, that X [itex]\subseteq[/itex] G but in my proof X may not be a subset of N.

Thanks in advance.

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# X-s-permutable subgroups.

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