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