Normally ζ-reversible profinite groups

Abstract

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 ζ-reversible. We conjecture that these groups are pronilpotent and we prove this conjecture if G is a normally ζ-reversible satisfying one of the following properties: G is prosoluble, G is perfect, all the nonabelian composition factors of G are alternating groups.