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A Theorem on Factorizable Groups of Odd Order

Published online by Cambridge University Press:  22 January 2016

Osamu Nagai
Osamu Nagai*
Affiliation:
Department of Mathematics, Yamaguchi University
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Recently, W. Feit [2] obtained some results on factorizable groups of odd order. By using his procedure and applying the theory of R. Brauer [1], we can prove the following theorem similar to that of W. Feit [2]:

Theorem. Let G be a factorizable group of odd order such that

G = HM

where H is a subgroup of order 3p, p being a prime greater than 3, and M is a maximal subgroup of G. Then G contains a proper normal subgroup which is contained either in H or in M.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 20 , June 1962 , pp. 201 - 203
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1962

References

[1] Brauer, R.: On permutation groups of prime degree and related classe of groups, Ann. of Math. 44, 44–57 (1943).Google Scholar
[2] Feit, W.: A theorem of factorizable groups, Proc. Amer. Math. Soc. 11, 11–658 (1960).Google Scholar
[3] Ikuta, T.: Über die Nichteinfachheit einer faktorisierbaren Gruppe, Nat. Sci. Rep. Lib. Arts Fac. Shizuoka Univ. 9, 9–1 (1956).Google Scholar
[4] Nagai, O.: On transitive groups that contain non-abelian regular subgroups, Osaka Math. J. 13, 13–199 (1961).Google Scholar
[5] Tuan, H.: On groups whose orders contain a prime number to the first power, Ann. of Math. 45, 45–110 (1944).Google Scholar
[6] Wielandt, H.: Vorlesung über Permutationsgruppen (Ausarbeitung von J. André.) Tübingen 1955.Google Scholar