The probability of commuting subgroups in arbitrary lattices of subgroups

A finite group $G$, in which two randomly chosen subgroups $H$ and $K$ commute, has been classified by Iwasawa in 1941. It is possible to define a probabilistic notion, which ``measures the distance'' of $G$ from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree $gsd(G)$ for two arbitrary sublattices $mathrm{S}(G)$ and $mathrm{T}(G)$ of the lattice of subgroups $mathrm{L}(G)$ of $G$. Upper and lower bounds for $gsd(G)$ are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and quotients, showing new numerical restrictions.