A subset X of a finite group G is said to be prime-power-independent if each element in X has prime power order and there is no proper subset Y of X with 〈Y,Φ(G)〉=〈X,Φ(G)〉, where Φ(G) is the Frattini subgroup of G. A group G is Bpp if all prime-power-independent generating sets for G have the same cardinality. We prove that, if G is Bpp, then G is solvable. Pivoting on some recent results of Krempa and Stocka (2014); Stocka (2020), this yields a complete classification of Bpp-groups.
Lucchini, A., Spiga, P. (2022). Independent sets of generators of prime power order. EXPOSITIONES MATHEMATICAE, 40(1), 140-154 [10.1016/j.exmath.2021.06.003].
Independent sets of generators of prime power order
Spiga P.
2022
Abstract
A subset X of a finite group G is said to be prime-power-independent if each element in X has prime power order and there is no proper subset Y of X with 〈Y,Φ(G)〉=〈X,Φ(G)〉, where Φ(G) is the Frattini subgroup of G. A group G is Bpp if all prime-power-independent generating sets for G have the same cardinality. We prove that, if G is Bpp, then G is solvable. Pivoting on some recent results of Krempa and Stocka (2014); Stocka (2020), this yields a complete classification of Bpp-groups.
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