Timeline for Centers of maximal subgroups in p-groups

4 events

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May 19, 2013 at 17:36	comment	added	Yassine Guerboussa		I think this example is due to A. Mann (with some modification, so be sure that any mistake in the above example is due to Y. Guerboussa)
May 19, 2013 at 17:28	comment	added	Yassine Guerboussa		To see this, pick two elements $x$ and $y$ of $K$ one in the center of $K$ (not in $Z(G)$) and the other not. In the vector space $F/H$, we can construct an automorphism which permutes $xH$ and $yH$, and which leaves $K/H$ invariant, this can be lifted to an automorphism of $K/U$ (since $G$ is relatively free), and so $Z(K)$ is not invariant under this automorphism, a contradiction. Perhaps, this may help in constructing other examples.
May 19, 2013 at 17:12	comment	added	Yassine Guerboussa		For any $n>2$ consider the relatively free group $G_n = F_n/U$, where $F_n$ is the free group on n-generators and $U = [F,F,F]F^p$. It follows that $Z(G)$ is equal to $H/U$ where $H = [F,F]F^p$ (since $H/U$ is the largest characteristic subgroup of $G_n)$. For any non-abelian subgroup $K$ of $G$ containing $Z(G)$, we have $Z(G)=Z(K)$ and so this hold for $K$ maximal.
May 18, 2013 at 13:36	history	asked	Yassine Guerboussa	CC BY-SA 3.0