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Given a finite group and an integer  q , we consider the poset of nilpotent subgroups of class less than  q  and its corresponding coset poset. These posets give rise to a family of finite Dirichlet series parameterized by the nilpotency class of the subgroups, which in turn reflect probabilistic and topological invariants determined by these subgroups. Connections of these series to filtrations of the classifying space of a group are discussed.

Higher commutativity and nilpotency in finite groups