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

Abstract

We give an elementary proof of the following remark: if G is a finite group and $\{g_1,\ldots ,g_d\}$ is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that $M\cap \{g_1,\ldots ,g_d\}=\varnothing .$ 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 $G=\langle g_1,\ldots ,g_d\rangle$ with $d(G)=d >3$ and with $\bigcap _{1\le i\le d}{{\mathrm{supp}}}(g_i)=\varnothing$? We obtain a weaker result in this direction: if $G=\langle g_1,\ldots ,g_d\rangle$ with $d(G)=d$, then ${{\mathrm{supp}}}(g_i)\cap {{\mathrm{supp}}}(g_j) \ne \varnothing$ for all $i, j\in \{1,\ldots ,d\}.$