Question on self-normalising $p$-subgroups of $p$-groups

Asked Oct 10 '22 at 14:03

Active Oct 10 '22 at 14:03

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Given a finite $p$-group $P$, we call a proper subgroup $Q<P$ centric if $C_P(Q)=Z(Q)$.

The question is: can a centric subgroup $Q$ be self-normalising, which means that $N_P(Q)=Q$?

I do not see any reason why not, and yet I am unable to find examples of self-normalising centric subgroups. So I am thinking that the answer is "no, centric implies that it is not self-normalising", but I also cannot prove that.

asked Oct 10 '22 at 14:03

FifteenPointOne

3

No proper subgroup of a finite $p$-group is self-normalizing. – Derek Holt Oct 10 '22 at 14:06
@DerekHolt ah, this I didn't know, I guess it explains it! Can you point me to a source? – FifteenPointOne Oct 10 '22 at 14:06
1

For one, it is handled in this very site. – Arturo Magidin Oct 10 '22 at 14:28

Question on self-normalising $p$-subgroups of $p$-groups

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