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

Maximal subgroups of finite groups avoiding the elements of a generating set We give an elementary proof of the following remark: if G is a finite group and $$\{g_1,\ldots ,g_d\}$$ { g 1 , … , g d } is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that $$M\cap \{g_1,\ldots ,g_d\}=\varnothing .$$ 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 $$G=\langle g_1,\ldots ,g_d\rangle$$ G = ⟨ g 1 , … , g d ⟩ with $$d(G)=d >3$$ d ( G ) = d > 3 and with $$\bigcap _{1\le i\le d}{{\mathrm{supp}}}(g_i)=\varnothing$$ ⋂ 1 ≤ i ≤ d supp ( g i ) = ∅ ? We obtain a weaker result in this direction: if $$G=\langle g_1,\ldots ,g_d\rangle$$ G = ⟨ g 1 , … , g d ⟩ with $$d(G)=d$$ d ( G ) = d , then $${{\mathrm{supp}}}(g_i)\cap {{\mathrm{supp}}}(g_j) \ne \varnothing$$ supp ( g i ) ∩ supp ( g j ) ≠ ∅ for all $$i, j\in \{1,\ldots ,d\}.$$ i , j ∈ { 1 , … , d } . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

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

, Volume 185 (3) – Oct 8, 2016
18 pages

/lp/springer_journal/maximal-subgroups-of-finite-groups-avoiding-the-elements-of-a-lleRr5x0X2
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-016-0985-y
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Monatshefte f�r MathematikSpringer Journals

Published: Oct 8, 2016

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