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- W4365440860 abstract "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 $xin G$, we denote by $P_{mathfrak{C}}(x,G)$ 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 $P_{mathfrak{C}}(x,G)>0$ for all $x in G.$ %Moreover we say that $G$ is $mathfrak C$-bounded-positive if there exists a positive constant $eta$ such that $P_{mathfrak{C}}(x,G)>eta$ for all $x in G.$ 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." @default.
- W4365440860 created "2023-04-15" @default.
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- W4365440860 date "2023-04-10" @default.
- W4365440860 modified "2023-09-28" @default.
- W4365440860 title "Probabilistic properties of profinite groups" @default.
- W4365440860 doi "https://doi.org/10.48550/arxiv.2304.04573" @default.
- W4365440860 hasPublicationYear "2023" @default.
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