Geodesic
Normally $\zeta$-reversible profinite groups
Algebra and discrete mathematics, Tome 21 (2016) no. 1, pp. 24-50.

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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.
Keywords: profinite groups, Dirichlet series. 
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Leone Cimetta; Andrea Lucchini. Normally $\zeta$-reversible profinite groups. Algebra and discrete mathematics, Tome 21 (2016) no. 1, pp. 24-50. http://geodesic.mathdoc.fr/item/ADM_2016_21_1_a3/

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