Maximal subgroups and PST-groups

Adolfo Ballester-Bolinches; James Beidleman; Ramón Esteban-Romero; Vicent Pérez-Calabuig

Open Mathematics (2013)

  • Volume: 11, Issue: 6, page 1078-1082
  • ISSN: 2391-5455

Abstract

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A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.

How to cite

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Adolfo Ballester-Bolinches, et al. "Maximal subgroups and PST-groups." Open Mathematics 11.6 (2013): 1078-1082. <http://eudml.org/doc/269271>.

@article{AdolfoBallester2013,
abstract = {A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.},
author = {Adolfo Ballester-Bolinches, James Beidleman, Ramón Esteban-Romero, Vicent Pérez-Calabuig},
journal = {Open Mathematics},
keywords = {Finite groups; Permutability; Sylow-permutability; Maximal subgroups; Supersolubility; finite groups; solvable T-groups; permutable subgroups; S-permutable subgroups; transitive permutability; transitive Sylow-permutability; maximal subgroups; supersolubility; finite T-groups; subnormal subgroups; transitive normality},
language = {eng},
number = {6},
pages = {1078-1082},
title = {Maximal subgroups and PST-groups},
url = {http://eudml.org/doc/269271},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Adolfo Ballester-Bolinches
AU - James Beidleman
AU - Ramón Esteban-Romero
AU - Vicent Pérez-Calabuig
TI - Maximal subgroups and PST-groups
JO - Open Mathematics
PY - 2013
VL - 11
IS - 6
SP - 1078
EP - 1082
AB - A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.
LA - eng
KW - Finite groups; Permutability; Sylow-permutability; Maximal subgroups; Supersolubility; finite groups; solvable T-groups; permutable subgroups; S-permutable subgroups; transitive permutability; transitive Sylow-permutability; maximal subgroups; supersolubility; finite T-groups; subnormal subgroups; transitive normality
UR - http://eudml.org/doc/269271
ER -

References

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  1. [1] Ballester-Bolinches A., Esteban-Romero R., Asaad M., Products of Finite Groups, de Gruyter Exp. Math., 53, Walter de Gruyter, Berlin, 2010 http://dx.doi.org/10.1515/9783110220612[Crossref] Zbl1206.20019
  2. [2] Ballester-Bolinches A., Ezquerro L.M., Classes of Finite Groups, Math. Appl. (Springer), 584, Springer, Dordrecht, 2006 Zbl1102.20016
  3. [3] Doerk K., Hawkes T., Finite Soluble Groups, de Gruyter Exp. Math., 4, Walter de Gruyter, Berlin, 1992 http://dx.doi.org/10.1515/9783110870138[Crossref] Zbl0753.20001
  4. [4] Gaschütz W., Über die Ø-Untergruppe endlicher Gruppen, Math. Z., 1953, 58, 160–170 http://dx.doi.org/10.1007/BF01174137[Crossref] Zbl0050.02202
  5. [5] Huppert B., Endliche Gruppen I, Grundlehren Math. Wiss., 134, Springer, Berlin, Berlin-New York, 1967 http://dx.doi.org/10.1007/978-3-642-64981-3[Crossref] 
  6. [6] Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25 http://dx.doi.org/10.1007/s00013-010-0207-0[WoS][Crossref] Zbl1230.20020

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