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Fixed Point Free Actions of Groups of Exponent 5

  • Enrico Jabara (a1)
Abstract

In this paper we prove that if V is a vector space over a field of positive characteristric p ≠ 5 then any regular subgroup A of exponent 5 of GL(V) is cyclic. As a consequence a conjecture of Gupta and Mazurov is proved to be true.

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References
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[1]Adyan, S. I., ‘Periodic groups of odd exponent’, in: Proceedings of the Second International Conference on the Theory of Groups, Australian Nat. Univ., 1973, Lecture Notes in Math. 372 (Springer, Berlin, 1974) pp. 812.
[2]Gupta, N. D. and Mazurov, V. D., ‘On groups with small orders of elements’, Bull. Austral. Math. Soc. 60 (1999), 197205.
[3]Hall, M., ‘Solution of the Burnside problem for exponent six’, Illinois J. Math. 2 (1958), 764786.
[4]Levi, F. W. and van der Waerden, B. L., ‘Über eine besondere Klasse von Gruppen’, Abh. Math. Sem. Univ. Hamburg 9 (1932), 154158.
[5]Mazurov, V. D., ‘Groups of exponent 60 with prescribed orders of elements’, Algebra i Logika 39 (2000), 329346; English translation: Algebra and Logic 39 (2000), 189198.
[6]Neumann, B. H., ‘Groups whose elements have bounded orders’, J. London Math. Soc. 12 (1937), 195198.
[7]Ol'šanskii, A. Ju., ‘An infinite group with subgroups of prime order’, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 309321.
[8]Robinson, D. J. S., A course in the theory of groups (Springer, Berlin, 1982).
[9]Sanov, I. N., ‘Solution of Burnside's problem for exponent 4’, Leningrad Univ. Ann. Math. Ser. 10 (1940), 166170.
[10]Zhurtov, A. K. and Mazurov, V. D., ‘A recognition of simple groups L2(2m) in the class of all groups’, Sibirsk. Math. Zh. 40 (1999), 7578;
English translation: Siberian Math. J. 40 (1999), 6264.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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