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 indices [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.
This is a joint work with Eloisa Detomi (University of Padova).
This video is part of the New York Group Theory Cooperative‘s group theory seminar series.
