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On the Structure of Frobenius Groups

Published online by Cambridge University Press:  20 November 2018

Walter Feit
Walter Feit*
Affiliation:
Cornell University
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Let G be a group which has a faithful representation as a transitive permutation group on m letters in which no permutation other than the identity leaves two letters unaltered, and there is at least one permutation leaving exactly one letter fixed. It is easily seen that if G has order mh, a necessary and sufficient condition for G to have such a representation is that G contains a subgroup H of order h which is its own normalizer in G and is disjoint from all its conjugates. Such a group G is called a Frobenius group of type (h, m).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 587 - 596
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Artin, E., Nesbitt, C. J. and Thrall, R. M., Rings with Minimum Condition (Ann Arbort, 1948).Google Scholar
2. Burnside, W., Theory of Groups of Finite Order (Dover, New York, 1955).Google Scholar
3. Feit, W., On a conjecture of Frobenius, Proc. Amer. Math. Soc, 7 (1956), 177–188.Google Scholar
4. Schmidt, O. J., Ueber die Frobenius Gruppen, C.R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 3–5.Google Scholar
5. Zassenhaus, H., The theory of Groups (Chelsea, New York, 1949).Google Scholar
6. Zassenhaus, H., Ueber endliche Fastkörper, Abh. Math. Sem. Hamburg Univ., 11 (1935), 187–220.Google Scholar
7. Zassenhaus, H., Ueber Liesche Ringe mit Primzahlcharacteristik,Abh. Math. Sem. Hamburg Univ., 13 (1940), 1–100.Google Scholar