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Some new permutability properties of hypercentrally embedded subgroups of finite groups

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

L. M. Ezquerro  and
X. Soler-Escrivà
L. M. Ezquerro
Affiliation:
Departamento de Matemática e Informática, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain, e-mail: ezquerro@unavarra.es
X. Soler-Escrivà
Affiliation:
Departament de Matemàtica Aplicada, Universitat d' Alacant, Campus de Sant Vicent, ap. Correus 99, 03080 Alacant, Spain, e-mail: xaro.soler@ua.es
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Abstract

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Hypercentrally embedded subgroups of finite groups can be characterized in terms of permutability as those subgroups which permute with all pronormal subgroups of the group. Despite that, in general, hypercentrally embedded subgroups do not permute with the intersection of pronormal subgroups, in this paper we prove that they permute with certain relevant types of subgroups which can be described as intersections of pronormal subgroups. We prove that hypercentrally embedded subgroups permute with subgroups of prefrattini type, which are intersections of maximal subgroups, and with F-normalizers, for a saturated formation F. In the soluble universe, F-normalizers can be described as intersection of some pronormal subgroups of the group.

Keywords

permutabilityfactorizations.

MSC classification

Secondary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20D30: Series and lattices of subgroups 20E15: Chains and lattices of subgroups, subnormal subgroups 20D35: Subnormal subgroups 20D40: Products of subgroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 2 , October 2005 , pp. 243 - 255
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Asaad, M. and Heliel, A. A., ‘On S-quasinormally embedded subgroups of finite groups’, J. Pure Appl. Algebra 165 (2001), 129–135.Google Scholar
[2]Ballester-Bolinches, A., ‘H-normalizers and local definitions of saturated formations of finite groups’, Israel J. Math. 67 (1989), 312–326.Google Scholar
[3]Ballester-Bolinches, A. and Ezquerro, L. M., ‘On maximal subgroups of finite groups’, Comm. Algebra 19 (1991), 2373–2394.CrossRefGoogle Scholar
[4]Ballester-Bolinches, A. and Ezquerro, L. M., ‘The Jordan-Hölder theorem and prefrattini subgroups of finite groups’, Glasgow Math. J. 37 (1995), 265–277.CrossRefGoogle Scholar
[5]Ballester-Bolinches, A. and Pedraza-Aguilera, M. C., ‘Sufficient conditions for supersolubility of finite groups’, J. Pure Appl. Algebra 127 (1998), 118–134.Google Scholar
[6]Carocca, A. and Maier, R., ‘Hypercentral embedding and pronormality’, Arch. Math. 71 (1998), 433–436.CrossRefGoogle Scholar
[7]Doerk, K. and Hawkes, T. O., Finite soluble groups (De Gruyter, Berlin, 1992).Google Scholar
[8]Ezquerro, L. M., Gómez-Fernández, M. and Soler-Escrivà, X., ‘On lattice properties of S-permutably embedded subgroups of finite soluble groups’, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (2005), 505–517.Google Scholar
[9]Ezquerro, L. M. and Soler-Escrivà, X., ‘Some factorizations involving hypercentrally embedded subgroups in finite soluble groups’, Proc. Groups St. Andrews-Oxford 2001 304 (2003), 190–196.Google Scholar
[10]Ezquerro, L. M. and Soler-Escrivà, X., ‘Some permutability properties related to F-hypercentrally embedded subgroups of finite groups’, J. Algebra 264 (2003), 279–295.Google Scholar
[11]Gaschütz, W., ‘Praefrattinigruppen’, Arch. Math. 13 (1962), 418–426.Google Scholar
[12]Hawkes, T. O., ‘Analogues of prefrattini subgroups’, in: Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Canberra (1965) (Gordon and Breach Science, 1967) pp. 145–150.Google Scholar
[13]Huppert, B. and Blackburn, N., Finite groups II (Springer, Berlin, 1982).CrossRefGoogle Scholar
[14]Kegel, O. H., ‘Sylow-Gruppen und Subnormalteiler endlicher Gruppen’, Math. Z. 78 (1962), 205–221.Google Scholar
[15]Makan, A., ‘Another characteristic conjugacy class of subgroups of finite soluble groups’, J. Austral. Math. Soc. 11 (1970), 395–400.Google Scholar
[16]Schmid, P., ‘Subgroups permutable with all Sylow subgroups’, J. Algebra 207 (1998), 285–293.Google Scholar
[17]Srinivasan, S., ‘Two sufficient conditions for supersolvability of finite groups’, Israel J. Math. 25 (1980), 210–214.CrossRefGoogle Scholar
[18]Cai, R. Yong, ‘Notes on π-quasi-normal subgroups in finite groups’, Proc. Amer. Math. Soc. 117 (1993), 631–636.Google Scholar