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A Class of Solvable Groups

Published online by Cambridge University Press:  20 November 2018

Daniel Gorenstein  and
I. N. Herstein
Daniel Gorenstein
Affiliation:
Clark UniversityandCornell University
I. N. Herstein
Affiliation:
Clark UniversityandCornell University
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Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB, where A and B are subgroups of G. The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B.

More recently the structure of finite groups G which have a representation in the form ABA, where A and B are subgroups of G, has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba', where a, a’ are in A, and b ≠ 1 is in B.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 311 - 320
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Gorenstein, D., A Class of Frobenius Groups, Can. J. Math., 11 (1959), 39–47.Google Scholar
2. Herstein, I. N. and Kaplansky, I., Groups of Cyclic Length Three, project document No. 13, Summer Mathematical Conference, Bowdoin College, Brunswick, Maine (1957).Google Scholar
3. Zassenhaus, H., The Theory of Groups, (New York, 1949).Google Scholar