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Finite group with Hall coverings

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Enrico Jabara  and
Maria Silvia Lucido
Enrico Jabara
Affiliation:
Dipartimento di Matematica Applicata, e Informatica, Università “Ca' Foscari” di Venezia, Via Torino 155, 31073 Venezia Mestre, Italy e-mail: [email protected]
Maria Silvia Lucido
Affiliation:
Dipartimento di Matematica, e Informatica, Università di Udine, Via delle Scienze 208, I-33100 Udine, Italy e-mail: [email protected]
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Abstract

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In this paper we describe the groups admitting a covering with Hall subgroups. We also determine the groups with a π1-Hall subgroup, where π1 is the connected component of the prime graph, containing the prime 2.

MSC classification

Secondary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20D25: Special subgroups (Frattini, Fitting, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 1 , February 2005 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Arad, Z. and Fisman, E., ‘On finite factorizable groups’, J. Algebra 86 (1984), 522–548.CrossRefGoogle Scholar
[2]Brandl, R., ‘Finite groups all of whose elements are of prime power order’, Boll. Un. Mat. Ital. A (5) 18 (1981), 491–493.Google Scholar
[3]Bubboloni, D. and Lucido, M. S., ‘Coverings of linear groups’, Comm. Algebra 30 (2002), 2143–2159.CrossRefGoogle Scholar
[4]Carter, R. W., Simple groups of Lie type (J. Wiley, London, 1972).Google Scholar
[5]Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R., Atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
[6]Dixon, J. D. and Mortimer, B., Permutation groups (Springer, New York, 1996).CrossRefGoogle Scholar
[7]Gorenstein, D., Finite groups (Harper & Row, New York, 1968).Google Scholar
[8]Gross, F., ‘Conjugacy of odd order Hall subgroups’, Bull. London Math. Soc. 19 (1987), 311–319.CrossRefGoogle Scholar
[9]Hall, P., ‘Theorems like Sylow's’, Proc. London Math. Soc. (3) 6 (1956), 286–304.CrossRefGoogle Scholar
[10]Higman, G., ‘Finite groups in which every element has prime power order’, J. London Math. Soc. 32 (1957), 335–342.CrossRefGoogle Scholar
[11]Higman, G., Odd characterisations of finite simple groups, Lectures at The University of Michigan (Summer semester, 1968).Google Scholar
[12]Huppert, B. and Blackburn, N., Finite groups III (Springer, Berlin, 1982).CrossRefGoogle Scholar
[13]Iiyori, N. and Yamaki, H., ‘Prime graph components of the simple groups of Lie type over the field of even characteristic’, J. Algebra 155 (1993), 335–343.CrossRefGoogle Scholar
[14]Kondrat'ev, A. S., ‘Prime graph components of finite simple groups’, Mat. Sb. 180 (1989), 787–797;Google Scholar
English translation: Math. of the USSR 67 (1990), 235–247.CrossRefGoogle Scholar
[15]Levchuk, V. M. and Nuzhin, Ya. N., ‘The structure of Ree groups’, Algebra i Logika 24 (1985), 26–41;Google Scholar
English translation: Algebra and Logic 24 (1985), 16–26.CrossRefGoogle Scholar
[16]Liebeck, M. W., ‘On the orders of maximal subgroups of the finite classical groups’, Proc. London Math. Soc. (3) 50 (1985), 426–446.CrossRefGoogle Scholar
[17]Liebeck, M. W. and Saxl, J., ‘On the orders of maximal subgroups of the finite exceptional groups of Lie type’, Proc. London Math. Soc. (3) 55 (1987), 299–330.CrossRefGoogle Scholar
[18]Lucido, M. S., ‘Prime graph components of finite almost simple groups’, Rend. Sem. Mat. Univ. Padova 102 (1999), 1–22.Google Scholar
(‘Addendum…’, Rend. Sem. Mat. Univ. Padova 107 (2002), 1–2.)Google Scholar
[19]Malle, G., ‘The maximal subgroups of 2 F4 (q2)’, J. Algebra 139 (1991), 52–69.CrossRefGoogle Scholar
[20]Malle, G., Saxl, J. and Weigel, T., ‘Generation of classical groups’, Geom. Dedicata 49 (1994), 85–116.CrossRefGoogle Scholar
[21]Martineau, P., ‘On 2-modular representations of the Suzuki groups’, Amer. J. Math. 94 (1972), 55–72.CrossRefGoogle Scholar
[22]Schmidt, R., Subgroup lattices of groups (De Gruyter, Berlin, 1994).CrossRefGoogle Scholar
[23]Spitznagel, E. L. Jr, ‘Hall subgroups of certain families of finite groups’, Math. Z. 97 (1967), 259–290.CrossRefGoogle Scholar
[24]Suzuki, M., ‘Finite groups with nilpotent centralizer’, Trans. Amer. Math. Soc. 99 (1961), 425–470.CrossRefGoogle Scholar
[25]Suzuki, M., ‘On a class of doubly transitive groups’, Ann. of Math. (2) 75 (1962), 105–145.CrossRefGoogle Scholar
[26]Williams, J. S., ‘Prime graph components of finite groups’, J. Algebra 69 (1981), 487–513.CrossRefGoogle Scholar
[27]Zacher, G., ‘Sull' ordine di un gruppo finito risolubile somma dei sottogruppi di sylow’, Atti Acc. Naz. Lincei (8) 20 (1956), 171–174.Google Scholar
[28]Zacher, G., ‘Sui gruppi finiti somma dei loro sottogruppi di Sylow’, Rend. Sem. Mat. Univ. Padova 27 (1957), 267–275.Google Scholar