Title: The probability of commuting subgroups in arbitrary lattices of subgroups
by Seid Kassaw (University of Cape Town) as part of Topological Groups

Lecture held in Elysium.

Abstract
The subgroup commutativity degree $sd(G)$ of a finite group $G$ was introduced
almost ten years ago and deals with the number of commuting subgroups in the
subgroup lattice $L(G)$ of $G$. The extremal case $sd(G) = 1$ detects a class of groups
classified by Iwasawa in 1941 (in fact, $sd(G)$ represents a probabilistic measure which
allows us to understand how far $G$ is from the groups of Iwasawa). This means
$sd(G) = 1$ if and only if $G$ is the direct product of its Sylow $p$-subgroups and these
are all modular; or equivalently $G$ is a nilpotent modular group. Therefore, $sd(G)$ is
strongly related to structural properties of $L(G)$ and $G$.

In this talk, we introduce a new notion of probability $gsd(G)$ in which two arbitrary sublattices $S(G)$ and $T(G)$ of $L(G)$ are involved simultaneously. In case
$S(G) = T(G) = L(G)$, we find exactly $sd(G)$. 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. We present the commutativity
and subgroup commutativity degree for infinite groups and put some open problems
for further generalization.