Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T07:52:28.134Z Has data issue: false hasContentIssue false

A note on minimal coverings of groups by subgroups

Published online by Cambridge University Press:  09 April 2009

R. A. Bryce
Affiliation:
School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia e-mail: [email protected]
L. Serena
Affiliation:
Dipartimento di Matematica, e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A cover for a group is a finite set of subgroups whose union is the whole group. A cover is minimal if its cardinality is minimal. Minimal covers of finite soluble groups are categorised; in particular all but at most one of their members are maximal subgroups. A characterisation is given of groups with minimal covers consisting of abelian subgroups.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bryce, R. A., Fedri, V. and Serena, L., ‘Covering groups with subgroups’, Bull. Austral. Math. Soc. 55 (1997), 469476.Google Scholar
[2]Bryce, R. A., Fedri, V. and Serena, L., ‘Subgroup coverings of some linear groups’, Bull. Austral. Math. Soc. 60 (1999), 227238.CrossRefGoogle Scholar
[3]Cohen, J. H. E., ‘On n-sum groups’, Math. Scand. 75 (1994), 4458.Google Scholar
[4]Doerk, K. and Hawkes, T., Finite soluble groups (Walter de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
[5]Gaschütz, W., ‘Existenz und Konjugiertsein von Untergruppen, die in endlichen auflösbaren Gruppen durch gewisse Indexschranken definiert sind’, J. Algebra 53 (1978), 120.CrossRefGoogle Scholar
[6]Kegel, O., ‘Die Nilpotenz der H p-Gruppen’, Math. Z. 75 (1960/1961), 373376.CrossRefGoogle Scholar
]7[Neumann, B. H., ‘Groups covered by finitely many cosets’, Publ. Math. Debrecen 3 (1954), 227242.Google Scholar
]8[Neumann, B. H., ‘A problem of Paul Erdös on groups’, J. Austral. Math. Soc. Ser. A 21 (1976), 467472.CrossRefGoogle Scholar
[9]Passman, D., Permutation groups (W. A. Benjamin, Inc., New York, 1968).Google Scholar
[10]Suzuki, M., ‘On a finite group with a partition’, Arch. Math. 12 (1961), 241254.CrossRefGoogle Scholar
[11]Tomkinson, M. J., ‘Hypercentre-by-finite groups’, Publ. Math. Debrecen 40 (1992), 313321.Google Scholar
[12]Tomkinson, M. J., ‘Groups as the union of proper subgroups’, Math. Scand. 81 (1997), 189198.CrossRefGoogle Scholar