Normally \(\zeta\)-reversible profinite groups

Leone Cimetta, Andrea Lucchini


We examine (finitely generated) profinite groups in which two formal Dirichlet series, the normal subgroup zeta function and the normal probabilistic zeta function, coincide; we call these groups normally \(\zeta\)-reversible. We conjecture that these groups are pronilpotent and we prove this conjecture if \(G\) is a normally \(\zeta\)-reversible satisfying one of the following properties: \(G\) is prosoluble, \(G\) is perfect, all the nonabelian composition factors of \(G\) are alternating groups.


profinite groups, Dirichlet series

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