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

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In summary: Hi, I just wanted to let you know that I've contacted the author and he told me that there was a mistake in his proof. He just altered his definition to make it work. There may be more things that need to be fixed, but I just wanted to let you know so that you're not wondering if everything is okay. Thanks for your help so far.
  • #1
moont14263
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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.
 
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  • #2
moont14263 said:
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
 
  • #3
Thanks for the advice. I'll send them an email.
 
  • #4
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.
 
  • #5
moont14263 said:
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
 

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

What are "X-s-permutable subgroups"?

"X-s-permutable subgroups" refer to subgroups in a group that commute with every subgroup in a certain class of subgroups, denoted by X. This means that every subgroup in X can be permuted with the subgroup without changing the structure of the group.

What is the significance of "X-s-permutable subgroups" in group theory?

"X-s-permutable subgroups" play an important role in studying the structure and properties of groups. They provide a way to classify groups and understand the relationships between subgroups. They also have applications in other areas of mathematics, such as in the study of finite groups and group actions.

How are "X-s-permutable subgroups" different from normal subgroups?

"X-s-permutable subgroups" are a generalization of normal subgroups. While normal subgroups commute with all subgroups in the group, "X-s-permutable subgroups" only commute with subgroups in a specific class X. This means that not all normal subgroups are "X-s-permutable", but all "X-s-permutable subgroups" are normal if X contains all subgroups of the group.

Can "X-s-permutable subgroups" exist in non-abelian groups?

Yes, "X-s-permutable subgroups" can exist in non-abelian groups. In fact, they are often used to study the structure of non-abelian groups and their subgroups. However, the class X must be carefully chosen to ensure that the definition of "X-s-permutable" is satisfied in a non-abelian group.

Are "X-s-permutable subgroups" always subnormal?

No, "X-s-permutable subgroups" are not always subnormal. A subgroup is subnormal if it has a chain of subgroups leading to the whole group, where each subgroup is normal in the next one. While all subnormal subgroups are "X-s-permutable", the converse is not necessarily true. There are examples of "X-s-permutable subgroups" that are not subnormal, even in abelian groups.

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