Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T18:17:29.558Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE NUMBER OF PRIME ORDER SUBGROUPS OF FINITE GROUPS

Part of: Generalizations Representation theory of groups

Published online by Cambridge University Press:  15 December 2009

TIMOTHY C. BURNESS  and
STUART D. SCOTT
TIMOTHY C. BURNESS*
Affiliation:
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK (email: [email protected])
STUART D. SCOTT
Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand (email: [email protected])
*
✉For correspondence; 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.

Let G be a finite group and let δ(G) be the number of prime order subgroups of G. We determine the groups G with the property δ(G)≥∣G∣/2−1, extending earlier work of C. T. C. Wall, and we use our classification to obtain new results on the generation of near-rings by units of prime order.

Keywords

finite groupsprime order subgroupsnear-rings

MSC classification

Secondary: 20D06: Simple groups 16Y30: Near-rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 3 , December 2009 , pp. 329 - 357
Copyright
Copyright © Australian Mathematical Publishing Association, Inc. 2009

References

[1]Burness, T. C., ‘Fixed point ratios in actions of finite classical groups, II’, J. Algebra 309 (2007), 80–138.Google Scholar
[2] The GAP Group, GAP  – Groups, Algorithms, and Programming, Version 4.4.Google Scholar
[3]Gorenstein, D., Finite Groups (Harper & Row, New York, 1968).Google Scholar
[4]Hegarty, P. V., ‘Soluble groups with an automorphism inverting many elements’, Math. Proc. R. Ir. Acad. 105A (2005), 59–73.CrossRefGoogle Scholar
[5]Kleidman, P. B. and Liebeck, M. W., The Subgroup Structure of the Finite Classical Groups, London Mathematical Society Lecture Note Series, 129 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[6]Lawther, R., Liebeck, M. W. and Seitz, G. M., ‘Fixed point ratios in actions of finite exceptional groups of Lie type’, Pacific J. Math. 205 (2002), 393–464.Google Scholar
[7]Liebeck, H. and MacHale, D., ‘Groups with automorphisms inverting most elements’, Math. Z. 124 (1972), 51–63.CrossRefGoogle Scholar
[8]Manning, W. A., ‘Groups in which a large number of operators may correspond to their inverses’, Trans. Amer. Math. Soc. 7 (1906), 233–240.Google Scholar
[9]Miller, G. A., ‘Non-abelian groups admitting more than half inverse correspondences’, Proc. Nat. Acad. Sci. 16 (1930), 168–172.CrossRefGoogle Scholar
[10]Neumaier, C., ‘The fraction of bijections generating the near-ring of 0-preserving functions’, Arch. Math. (Basel) 82 (2005), 497–507.Google Scholar
[11]Neumann, H., ‘Varieties of groups and their associated near-rings’, Math. Z. 65 (1956), 36–69.CrossRefGoogle Scholar
[12]Pilz, G., Near-Rings, North-Holland Mathematics Studies, 23 (North-Holland Publishing Co., Amsterdam, 1983).Google Scholar
[13]Potter, W. M., ‘Nonsolvable groups with an automorphism inverting many elements’, Arch. Math. (Basel) 50 (1988), 292–299.Google Scholar
[14]Scott, S. D., Generators of Finite Transformation Nearrings, Book in preparation.Google Scholar
[15]Scott, S. D., ‘Involution near-rings’, Proc. Edinb. Math. Soc. 22 (1979), 241–245.Google Scholar
[16]Scott, S. D., ‘Transformation near-rings generated by a unit of order three’, Algebra Colloq. 4 (1997), 371–392.Google Scholar
[17]Vaughan-Lee, M., The Restricted Burnside Problem, London Mathematical Society Monographs, 8 (Oxford University Press, Oxford, 1993).CrossRefGoogle Scholar
[18]Wall, C. T. C., ‘On groups consisting mostly of involutions’, Math. Proc. Cambridge Philos. Soc. 67 (1970), 251–262.CrossRefGoogle Scholar