Maximal subgroups of finite groups avoiding the elements of a generating set

• Published in: Monatshefte für Mathematik Vol. 185; no. 3; pp. 455 - 472
• Main Authors: Lucchini, Andrea, Spiga, Pablo
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.03.2018; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; Permutations; Subgroups; Group generation; Permutation groups; Primary 20E28; 20F05; Primitive groups; Maximal subgroups; Secondary 20B15
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-016-0985-y

We give an elementary proof of the following remark: if G is a finite group and { g 1 , … , g d } is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that M ∩ { g 1 , … , g d } = ∅ . 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 = ⟨ g 1 , … , g d ⟩ with d ( G ) = d > 3 and with ⋂ 1 ≤ i ≤ d supp ( g i ) = ∅ ? We obtain a weaker result in this direction: if G = ⟨ g 1 , … , g d ⟩ with d ( G ) = d , then supp ( g i ) ∩ supp ( g j ) ≠ ∅ for all i , j ∈ { 1 , … , d } .