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A Classification Of n-Abelian Groups

Published online by Cambridge University Press:  20 November 2018

J. L. Alperin
J. L. Alperin*
Affiliation:
Northeastern University, Boston, Massachusetts University of Chicago, Chicago, Illinois
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The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,

(xy)n = xnyn

for all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.

Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1238 - 1244
Copyright
Copyright © Canadian Mathematical Society 1969

References

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