A note on finite groups with the maximal permutiser condition

  • Adolfo Ballester-BolinchesEmail author
  • John Cossey
  • ShouHong Qiao
Original Paper


A finite group G is said to satisfy the maximal permutiser condition, or G is an MPC-group, if for any maximal subgroup M of G, there is an element \(x\in G{\setminus }M\) such that \(G=M\langle x\rangle \). In this note, we show that the class of MPC-groups is not residually closed and so it is not a formation. It answers a question posed in Qiao et al. (J Algebra Appl 12(5):1250217, 2013). Following Ballester-Bolinches and Esteban-Romero (Commun Algebra 30(12):5757–5770, 2002), a finite group G is said to be a QP-group if G is soluble and if F is a non-cyclic chief factor of G, then F has order 4 and G induces the full automorphism group in F. We prove that the class of all QP-groups is the unique largest formation contained in the class of all MPC-groups. A detailed description of the MPC-groups is also given.


Finite group Soluble group Permutability Formations 

Mathematics Subject Classification

20D10 20D15 



The first author was supported by Proyecto MTM2010-19938-C03-01 from Department of I + D+ i of MEC (Spain), and Project from the National Natural Science Foundation of China (NSFC, No. 11271085). The third author was supported by Project from the National Natural Science Foundation of China (NSFC, No. 11201082), Cultivation Program for Outstanding Young College Teachers (Yq2013061) of Guangdong Province and Pei Ying Yu Cai Project of GDUT.


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Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  • Adolfo Ballester-Bolinches
    • 1
    Email author
  • John Cossey
    • 2
  • ShouHong Qiao
    • 3
  1. 1.Departament d’ÀlgebraUniversitat de ValènciaValènciaSpain
  2. 2.Mathematics Department, School of Mathematical SciencesThe Australian National UniversityCanberraAustralia
  3. 3.School of Applied MathematicsGuangdong University of TechnologyGuangzhouPeople’s Republic of China

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