Subnormalizers and solvability in finite groups

DOI10.1016/J.JALGEBRA.2022.04.019arXiv2007.03554MaRDI QIDQ6344607

Pietro Gheri

Publication date: 7 July 2020

Abstract: For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,yinG$, the cyclic subgroup $langlexangle$ is subnormal in the subgroup $langlex,yangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G)leq1/6$ for every nonsolvable group $G$.

Mathematics Subject Classification ID

Finite solvable groups, theory of formations, Schunck classes, Fitting classes, (pi)-length, ranks (20D10) Sylow subgroups, Sylow properties, (pi)-groups, (pi)-structure (20D20) Subnormal subgroups of abstract finite groups (20D35) Probabilistic methods in group theory (20P05)

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