Recognizing the Non-Frattini Abelian Chief Factors of a Finite Group from Its Probabilistic Zeta Function

Patassini, Massimiliano

doi:10.1080/00927872.2011.610075

Cited by 2 publications

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“…The first summand P (p) G (s) collects the contribution given by the subgroups H of X such that H contains a Sylow p-subgroup, which are intransitive subgroups of G (see [10,Lemma 9]). In [10, Proposition 12] an explicit formula for the Dirichlet polynomial P (p) G (s) is given.…”

Section: Introductionmentioning

confidence: 99%

On the (non-)Contractibility of the Order Complex of the Coset Poset of an Alternating Group

Patassini¹

2013

Rend. Sem. Mat. Univ. Padova

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Let Alt k be the alternating group of degree k. In this paper we prove that the order complex of the coset poset of Alt k is non-contractible for a big family of k P N, including the numbers of the form k p m where m P f3; . . . ; 35g and p > k=2. In order to prove this result, we show that P G ( À 1) does not vanish, where P G (s) is the Dirichlet polynomial associated to the group G. Moreover, we extend the result to some monolithic primitive groups whose socle is a direct product of alternating groups.MATHEMATICS SUBJECT CLASSIFICATION (2010). 20D30, 20P05, 11M41. defined inductively by m G (G) 1, m G (H) À P K>H m G (K). The counterpart of the Dirichlet polynomial is called the probabilistic zeta function of G (see [1] and [6]).In particular, Brown ([2], § 3) showed that P G ( À 1) Àx(g(G)):It is a well-known fact that if D(g(G)) is contractible, then its reduced Euler characteristicx(g(G)) is zero. Hence, if P G ( À 1) T 0, then the simplicial complex associated to the group G is non-contractible. Moreover, in [2], Brown conjectured the following. CONJECTURE 1. If G is a finite group, then P G ( À 1) T 0. Hence the order complex of the coset poset of G is non-contractible.On the (non-)Contractibility of the Order Complex etc.

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Section: Introductionmentioning

confidence: 99%