Bases of certain finite groups
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 \(\Phi(G) =1\).
scientific article; zbMATH DE number 784557
scientific article