Subgroup commutativity degrees of finite groups. (Q1028303)

Let \(G\) be a finite group and let \(L(G)\) be the set of subgroups of \(G\). The author defines the subgroup commutativity degree of \(G\) by \(\text{sd}(G)=|L(G)|^{-2}|\{(H,K)\in L(G)^2\mid HK=KH\}|\). Clearly, \(\text{sd}(G)\) is the probability that two subgroups of \(G\) permute. The author states some simple general properties of \(\text{sd}(G)\). Then he gives a lower bound for \(\text{sd}(G)\) in terms of \(\text{sd}(N)\), \(\text{sd}(G/N)\), \(|L(N)|\), and \(|L(G/N)|\) if \(N\) is a normal subgroup of \(G\). Finally, he computes the subgroup commutativity degrees of the finite dihedral groups and uses this result to determine \(\text{sd}(G)\) for all \(2\)-groups \(G\) with a cyclic maximal subgroup. For \(p>2\), any two subgroups of such a \(p\)-group \(G\) permute and hence \(\text{sd}(G)=1\).