TY - JOUR

T1 - A note on finite groups with the maximal permutiser condition

AU - Ballester-Bolinches, Adolfo

AU - Cossey, John

AU - Qiao, Shou Hong

N1 - Publisher Copyright:
© 2015, Springer-Verlag Italia.

PY - 2016/3/1

Y1 - 2016/3/1

N2 - 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 (Formula presented.) such that (Formula presented.). 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.

AB - 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 (Formula presented.) such that (Formula presented.). 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.

KW - Finite group

KW - Formations

KW - Permutability

KW - Soluble group

UR - http://www.scopus.com/inward/record.url?scp=84958757590&partnerID=8YFLogxK

U2 - 10.1007/s13398-015-0232-8

DO - 10.1007/s13398-015-0232-8

M3 - Article

SN - 1578-7303

VL - 110

SP - 247

EP - 250

JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

IS - 1

ER -