Is H X-s-permutable in N Given X May Not Be a Subset of N?

[moont14263]

I want to prove Lemma 2.1(1) in this paper, the first pdf file in the page
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.

 

[DonAntonio]

I want to prove Lemma 2.1(1) in this paper, the first pdf file in the page
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.

Indeed. Then either this is a condition we can wave in this case, or else we must take [itex]X\cap N[/itex] , which automatically would

make, apparently, the proof way harder as that [itex]x\in X[/itex] may well not be in N.

Write the authors an email asking them about this. My experience is that most of them (even very well known and famous authors) are

pretty nice and open to answer back when asked about something in their work.

DonAntonio

 

[moont14263]

Thanks for the advice. I'll send them an email.

 

[moont14263]

I contacted one of the authors and he told me that there was a mistake. He just altered his definition to make things work. I do not know if there are more things that need to be fixed. I just wrote this comment to let you know. Thank you very much for every one specially DonAntonio. As you said, he was a nice guy.

 

[DonAntonio]

I contacted one of the authors and he told me that there was a mistake. He just altered his definition to make things work. I do not know if there are more things that need to be fixed. I just wrote this comment to let you know. Thank you very much for every one specially DonAntonio. As you said, he was a nice guy.

I'm happy for you. It was expected that guy was a nice one: we mathematicians are lovely and lovable.

DonAntonio