Connected components of the category of elementary abelian $p$-subgroups.

Mazza, Nadia (2008) Connected components of the category of elementary abelian $p$-subgroups. Journal of Algebra, 320 (12). pp. 4242-4248. ISSN 0021-8693

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Abstract

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank $2$ in a finite $p$-group $G$, for an odd prime $p$. Namely, it is $p$ if $G$ has rank at least $3$ and it is $p+1$ if $G$ has rank $2$. More precisely, if $G$ has rank $2$, there are exactly $1,~2,~p+1$, or possibly $3$ classes for some $3$-groups of maximal nilpotency class.