Sets of prime power order generators of finite groups

A. Stocka

Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 129-138

Résumé

A subset $X$ of prime power order elements of a finite group $G$ is called $\mathrm{pp}$-independent if there is no proper subset $Y$ of $X$ such that $\langle Y,\Phi(G) \rangle = \langle X,\Phi(G) \rangle$, where $\Phi(G)$ is the Frattini subgroup of $G$. A group $G$ has property $\mathcal{B}_{pp}$ if all $\mathrm{pp}$-independent generating sets of $G$ have the same size. $G$ has the $\mathrm{pp}$-basis exchange property if for any $\mathrm{pp}$-independent generating sets $B_1, B_2$ of $G$ and $x\in B_1$ there exists $y\in B_2$ such that $(B_1\setminus \{x\})\cup \{y\}$ is a $\mathrm{pp}$-independent generating set of $G$. In this paper we describe all finite solvable groups with property $\mathcal{B}_{pp}$ and all finite solvable groups with the $\mathrm{pp}$-basis exchange property.

Détail

Keywords: finite groups, independent sets, minimal generating sets, Burnside basis theorem.