Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T02:26:42.274Z Has data issue: false hasContentIssue false

On groups’ with small orders of elements

Published online by Cambridge University Press:  17 April 2009

Narain D. Gupta
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Canada, e-mail: [email protected]
Victor D. Mazurov
Affiliation:
Institute of Mathematics, Novosibirsk 630090, Russia, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

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.

For a periodic group G, denote by ω(G) the set of orders of elements in G. We prove that if ω(G) is a proper subset of the set {1, 2, 3, 4, 5} then either G is locally finite or G contains a nilpotent normal subgroup N such that G/N is a 5-group.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Hall, M. Jr, ‘Solution of the Burnside problem for exponent six’, Illinois J. Math. 2 (1958), 764786.CrossRefGoogle Scholar
[2]Higman, G., ‘Groups and rings having automorphisms without non-trivial fixed elements’, J. London Math. Soc. 32 (1957), 321334.CrossRefGoogle Scholar
[3]Huppert, B., Endliche Gruppen I (Springer Verlag, Berlin, Heidelberg, New York, 1979).Google Scholar
[4]Levi, F.W., ‘Groups in which the commutator operation satisfies certain algebraical conditions’, J. Indian Math. Soc. 6 (1942), 8797.Google Scholar
[5]Levi, F., van der Waerden, B.L., ‘Über eine besondere Klasse von Gruppen’, Abh. Math. Sem. Univ. Hamburg 9 (1932), 154158.CrossRefGoogle Scholar
[6]Neumann, B.H., ‘Groups whose elements have bounded orders’, J. London Math. Soc. 12 (1937), 195198.CrossRefGoogle Scholar
[7]Neumann, B.H., ‘Groups with automorphisms that leave only the neutral element fixed’, Arch. Math. 7 (1956), 15.CrossRefGoogle Scholar
[8]Sanov, I.N., ‘Solution of Burnside's problem for exponent 4’, (Russian), Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 10 (1940), 166170.Google Scholar
[9]Schönert, M., et al. , Groups, algorithms and programming (Lehrstuhl D für Mathematik, RWTH Aachen, 1993).Google Scholar
[10]Shunkov, V.P., ‘On periodic groups with an almost regular involution’, (Russian), Algebra i Logika 11 (1972), 470493. Algebra and Logic 11 (1972), 260–272.Google Scholar
[11]Zhurtov, A.K. and Mazurov, V.D., ‘A recognition of simple groups L 2 (2m) in the class of all groups’, (Russian), Sibirsk. Mat. Zh. 40 (1999), 7578. Siberian Math. J. 40 (1999), 62–64.Google Scholar