Bases of certain finite groups
type: Item
instance of: scholarly article
title: Bases of certain finite groups
zbMATH Open document ID: 0831.20039
author: Libero Verardi
author: Raffaele Scapellato
published in: Annales Mathématiques Blaise Pascal
publication date: 1996-02-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79107
review text: Let \(G\) be a finite group, and let \(\Phi(G)\) be its Frattini subgroup. Consider the mapping \(c: P(G)\to P(G)\) defined by \(c(X)=\langle X,\Phi(G)\rangle\). A subset of \(X\) of \(G\) is said to be independent (hierarchical) if \(c(Y)\neq c(X)\) for all \(Y\subset X\) (resp. if its elements can be ordered in a sequence \(x_1, x_2,\dots, x_m\) such that \(c(x_1, x_2,\dots, x_{k -1})\neq c(x_1, x_2,\dots, x_k)\) for all \(k=2,\dots, m\), and \(c(\emptyset)\neq c(x_1))\). The group \(G\) is said to be a matroid group if it satisfies the following two axioms: (M1) the minimal generating sets of \(G\) are exactly the independent subsets \(X\) of \(G\) with \(c(X)=G\); (M2) each independent subset \(X\) of \(G\) is contained in a minimal generating set of \(G\). The authors [Boll. Unione Mat. Ital., VII. Ser., A5, 187-194 (1991; Zbl 0788.20019) and Ann. Sci. Univ. Blaise Pascal Clermont-Ferrand II 95, Math. 26, 51-60 (1990; Zbl 0788.20018)] described these groups. In particular, all matroid groups are solvable and all finite \(p\)-groups are matroid groups. In the present paper the authors prove the following two theorems. Theorem 1: If \(G\) is solvable, and \(H=G/\Phi(G)\), then \(G\) satisfies (M2) if and only if \(H\) is abelian, or the Fitting subgroup \(F(H)\) of \(H\) has prime index in \(H\) and all its subgroups are normal in \(H\). Theorem 2: All hierarchical generating sets of \(G\) are minimal generating sets of \(G\) if and only if \(G\) is a matroid group with \(\Phi(G) =1\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 784557
zbMATH Keywords: solvable groups
zbMATH Keywords: finite groups
zbMATH Keywords: Frattini subgroup
zbMATH Keywords: minimal generating sets
zbMATH Keywords: independent subsets
zbMATH Keywords: matroid groups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Fitting subgroup
zbMATH Keywords: hierarchical generating sets
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
OpenAlex ID: W2325500646
cites work: Une Classe Particuliere De Matroides Parfaits
cites work: Q3973544
cites work: Q3757929
cites work: The connectivity of hierarchical Cayley digraphs
cites work: Basis properties for inverse semigroups
cites work: Q3135082
cites work: Q3984382
cites work: Q3974643
cites work: Q5331572
DOI: 10.5802/AMBP.14
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description: scientific article; zbMATH DE number 784557