On two Möbius functions for a finite non-solvable group

Volta, Francesca Dalla; Zini, Giovanni

doi:10.1080/00927872.2021.1924184

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2023

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“…holds for any H ≤ G whenever G is finite and solvable. Following [4], we say that G satisfies the (µ, λ)-property if (1.2) holds for any H ≤ G. Several classes of non-solvable groups satisfy the (µ, λ)-property, for example all the minimal non-solvable groups (see [4]). However it is known that the (µ, λ)-property does not hold for every finite group.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

The A-Möbius function of a finite group

Volta¹,

Lucchini²

2023

Ars Math. Contemp.

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The Möbius function of the subgroup lattice of a finite group G has been introduced by Hall and applied to investigate several different questions. We propose the following generalization. Let A be a subgroup of the automorphism group Aut(G) of a finite group G and denote by C A (G) the set of A-conjugacy classes of subgroups of G. For H ≤ G let [H] A = { H a | a ∈ A} be the element of C A (G) containing H. We may define an ordering in C A (G) in the following way: [H] A ≤ [K] A if H a ≤ K for some a ∈ A. We consider the Möbius function µ A of the corresponding poset and analyse its properties and possible applications.

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Section: Accepted Manuscriptmentioning

confidence: 99%