We obtain some results on the subgroups of finite solvable groups with non-zero Mobius function. These are used to prove a conjecture of Mann in the particular case of prosolvable groups: if G is a finitely generated prosolvable group, then the infinite ∑ μ(H,G)/|G:H|^s , where H ranges over all open subgroups, is absolutely convergent in some right half-plane of the complex plane.
Subgroups of solvable groups with non-zero Moebius function
LUCCHINI, ANDREA
2007
Abstract
We obtain some results on the subgroups of finite solvable groups with non-zero Mobius function. These are used to prove a conjecture of Mann in the particular case of prosolvable groups: if G is a finitely generated prosolvable group, then the infinite ∑ μ(H,G)/|G:H|^s , where H ranges over all open subgroups, is absolutely convergent in some right half-plane of the complex plane.
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