Departamento de Matemáticas UAM

Seminario Teoría de grupos 

28 DE MARZO DE 2023

11:30 Aula Roja, IFT

Speaker: Pavel Shumyatsky (University of Brasilia)

Title: Commuting probability for subgroups of a finite group

Abstract: If $K$ is a subgroup of a finite group $G$, the probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. The probability that two randomly chosen elements of $G$ commute is denoted by $Pr(G)$. A well known theorem, due to P. M. Neumann, says that if $G$ is a finite group such that $Pr(G)geqepsilon>0$, then $G$ has a normal subgroup $T$ such that the index $[G:T]$ and the order $|[T,T]|$ are both $epsilon$-bounded.

In the talk we will discuss a stronger version of Neumann's theorem: if $K$ is a subgroup of $G$ such that $Pr(K,G)geqepsilon$, then there is a normal subgroup $Tleq G$ and a subgroup $Bleq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $epsilon$-bounded.

This is a joint work with Eloisa Detomi (University of Padova).