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This article is cited in 1 scientific paper (total in 1 paper)
RESEARCH ARTICLE
Sets of prime power order generators of finite groups
A. Stocka Faculty of Mathematics University of Białystok K. Ciołkowskiego 1M 15-245 Białystok
Abstract:
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.
Keywords:
finite groups, independent sets, minimal generating sets, Burnside basis theorem.
Received: 17.10.2019 Revised: 17.12.2019
Citation:
A. Stocka, “Sets of prime power order generators of finite groups”, Algebra Discrete Math., 29:1 (2020), 129–138
Linking options:
https://www.mathnet.ru/eng/adm745 https://www.mathnet.ru/eng/adm/v29/i1/p129
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