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 ۱۹۴۱‎. ‎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.

Keywords:

‎‎Subgroup commutativity degree , Dihedral groups , Sublattices , Elementary abelian $p$-groups , polynomial functions