Seminar

Group Theory Seminar

Commuting probability for subgroups of a finite group

Speaker:  Pavel Shumyatsky (Universidad de Brasilia / Universidad del País Vasco)
Date:  Tuesday, 28 March 2023 - 11:30
Place:  Aula Roja, ICMAT

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 \(Pr(G)\geq\epsilon>0\), 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 \(Pr(K,G)\geq\epsilon\), then there is a normal subgroup \(T\leq G\) and a subgroup \(B\leq K\) 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).