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.

Independent sets of generators of prime power order

Lucchini A.;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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3444188
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 1
  • OpenAlex ND
social impact