Geodesic
Sets of prime power order generators of finite groups
Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 129-138

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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. 
A. Stocka. Sets of prime power order generators of finite groups. Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 129-138. http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a11/
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