Tag - Probabilistic group theory

Pavel Shumyatsky: Commuting probability for subgroups of a finite group

7th April, 2022

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) ≥ ε, then G has a nilpotent normal subgroup T of class at most 2 such that both the index [G:T] and the order |[T,T]| are ε-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) ≥ ε, then there is a normal subgroup T ≤ G and a subgroup B ≤ K such that the indexes [G:T] and [K:B] and the order of the commutator subgroup [T,B] are ε-bounded.

We will also discuss a number of corollaries of this result. A typical application is that if in the above theorem K is the generalized Fitting subgroup F*(G), then G has a class-2-nilpotent normal subgroup R such that both the index [G:R] and the order of the commutator subgroup [R,R] are ε-bounded.

Victor Guerassimov: Random walks on groups. An introduction.

10th March, 2022

Geometric methods proved to be useful in the study of some groups. However the geometry of the Cayley graph of a group is rather different from the geometry of classical geometric objects such as homogeneous spaces of Lie groups. The similarity between these two geometries grows as the scale of observation increases. And the asymptototic behavior of them shows surprising similarity. Random walks is an essential tool in studying large-scale geometry of groups. On the other hand it is an interesting object for probabilists since many properties of general stochastic processes are manifested here in a rather simple form. In my talk, I will provide an elementary introduction to this vast area. No special knowledge beyond the usual university mathematics is required.