2023/04/10 by Detomi, Eloisa, Lucchini, Andrea, Morigi, Marta +1
#05C25 #20E18 #20F19 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2304.04573
Let \mathfrak C be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group G and an element x∈ G, we denote by P_\mathfrakC(x,G) the probability that x and a randomly chosen element of G generate a pro-\mathfrak C subgroup. We say that a profinite group G is \mathfrak C-positive if P_\mathfrakC(x,G)>0 for all x ∈ G. %Moreover we say that G is \mathfrak C-bounded-positive if there exists a positive constant η such that P_\mathfrakC(x,G)>η for all x ∈ G. We establish several equivalent conditions for a profinite group to be \mathfrak C-positive when \mathfrak C is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite \mathfrak C-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.