Seid Kassaw (University of Cape Town)

November 24, 2020 @ 6:00 am – 8:00 am
Lecture held in Elysium

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

Interactive livestream:
Lecture held in Elysium.

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.