We give an elementary proof of the following remark: if G is a finite group and {g1,…,gd} is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that M∩{g1,…,gd}=∅. This result leads us to investigate the freedom that one has in the choice of the maximal subgroup M of G. We obtain information in this direction in the case when G is soluble, describing for example the structure of G when there is a unique choice for M. When G is a primitive permutation group one can ask whether is it possible to choose in the role of M a point-stabilizer. We give a positive answer when G is a 3-generated primitive permutation group but we leave open the following question: does there exist a (soluble) primitive permutation group G=⟨g1,…,gd⟩ with d(G)=d>3 and with ⋂1≤i≤dsupp(gi)=∅? We obtain a weaker result in this direction: if G=⟨g1,…,gd⟩ with d(G)=d, then supp(gi)∩supp(gj)≠∅ for all i,j∈{1,…,d}
Lucchini, A., & SPIGA, P. (2018). Maximal subgroups of finite groups avoiding the elements of a generating set. MONATSHEFTE FÜR MATHEMATIK, 185(3), 455-472 [10.1007/s00605-016-0985-y].
Citazione: | Lucchini, A., & SPIGA, P. (2018). Maximal subgroups of finite groups avoiding the elements of a generating set. MONATSHEFTE FÜR MATHEMATIK, 185(3), 455-472 [10.1007/s00605-016-0985-y]. | |
Tipo: | Articolo in rivista - Articolo scientifico | |
Carattere della pubblicazione: | Scientifica | |
Presenza di un coautore afferente ad Istituzioni straniere: | No | |
Titolo: | Maximal subgroups of finite groups avoiding the elements of a generating set | |
Autori: | Lucchini, A; SPIGA, P | |
Autori: | SPIGA, PABLO (Ultimo) | |
Data di pubblicazione: | 2018 | |
Lingua: | English | |
Rivista: | MONATSHEFTE FÜR MATHEMATIK | |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s00605-016-0985-y | |
Appare nelle tipologie: | 01 - Articolo su rivista |
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