The subgroup commutativity degree of finite \(P\)-groups. (Q2786575)

The author introduced the subgroup commutativity degree NEWLINE\[NEWLINEsd(G)=\frac{|\{(H,K)\in L(G)\times L(G)\mid HK=KH\}|}{|L(G)|^2}NEWLINE\]NEWLINE of a finite group \(G\) [in J. Algebra 321, No. 9, 2508-2520 (2009; Zbl 1196.20024)]. Note that \(sd(G)=1\) if and only if each subgroup of \(G\) is permutable. On the other hand, the groups in which all subgroups are permutable are well known and called quasihamiltonian; they have been classified by K. Iwasawa long time ago and play a fundamental role in the general theory of lattices of subgroups. Roughly speaking, \(sd(G)\) describes the probability of finding quasihamiltonian groups among all finite groups.NEWLINENEWLINE The main result (see Theorem 1.1) of this short and elegant note deals with the description of a family of groups \(G_n\) such that \(\lim_{n\to\infty}sd(G_n)=0\). Actually the author shows that the same family of \(P\)-groups, involved in the original classification of Iwasawa, has this nice asymptotic property. The proof uses some combinatorial arguments, based on the number of subgroups of finite elementary abelian groups, which appear in the structure of \(P\)-groups.