Let G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, whenever G is solvable, the property (Formula presented.) holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M 12 being a counterexample. In this paper we investigate the relation between μ and λ for some classes of non-solvable groups, among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.
On two Möbius functions for a finite non-solvable group / Dalla Volta, F.; Zini, G.. - In: COMMUNICATIONS IN ALGEBRA. - ISSN 0092-7872. - 49:11(2021), pp. 4565-4576. [10.1080/00927872.2021.1924184]
On two Möbius functions for a finite non-solvable group
Zini G.
2021
Abstract
Let G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, whenever G is solvable, the property (Formula presented.) holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M 12 being a counterexample. In this paper we investigate the relation between μ and λ for some classes of non-solvable groups, among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.File | Dimensione | Formato | |
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