On the Poset of Conjugacy Classes of Subgroups of Finite p-Groups

For a finite group $G$, let ${cal C}(G)$ denote the poset of conjugacy classes $[S]$ of subgroups $S$ of $G$. The ordering is $[S_1]leq[S_2]$ iff $S_1leq S_2^g$ for some $gin G$. Let $H$ be a finite group such that ${cal C}(G)$ and ${cal C}(H)$ are order-isomorphic. Let $G$ is a $p$-group. It is known [{it R. Brandl}, Commun. Algebra 20, No. 10, 3043-3054 (1992; Zbl 0767.20008)] that $H$ is a $p$-group. Moreover, if $G$ is abelian or metacyclic, then $Gcong H$ [see {it L. Verardi} and the reviewer, Glasg. J. Math. 35, No. 3, 339-344 (1993; Zbl 0846.20020)].par Let $arphicolon{cal C}(G) o{cal C}(G)$ be an order-isomorphism. It is shown in this interesting paper that $arphi$ maps (conjugacy classes of) normal subgroups of $G$ onto normal subgroups of $H$. This answers a question of L. Verardi and the reviewer. Indeed, a stronger result is proved concerning $p$-groups $G$ with a $p$-group $U$ of operators. The specialization $U=1$ yields, for example, the known result that a projectivity between two $p$-groups maps some principal series of $G$ onto a principal series of $H$. Another consequence of the main theorem is that the classes of $p$-groups of maximal nilpotency class, and of powerful $p$-groups are determined by their posets of subgroups. Moreover, if $G$ is a modular $p$-group, then $Gcong H$.par Generally speaking, the paper supports the philosophy that posets of subgroups, although very much smaller, behave much better than the lattices of all subgroups, but seem to contain enough information to recover many properties of the group under consideration. [R.Brandl (Würzburg)]