2021/02/13 by Detomi, Eloisa, Shumyatsky, Pavel · 2 citations
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2102.06983
Let K be 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). Assume that Pr(K,G)≥ε for some fixed ε>0. We show that 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. This extends the well known theorem, due to P. M. Neumann, that covers the case where K=G. We deduce a number of corollaries of this result. A typical application is that if 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. In the same spirit we consider the cases where K is a term of the lower central series of G, or a Sylow subgroup, etc.