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A New Type of Characteristic Subgroup of Prime-Power Groups

Published online by Cambridge University Press:  20 November 2018

H. R. Brahana*
Affiliation:
University of Illinois
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In a recent paper (1) the fifty-eight metabelian groups of order p11 that are generated by five elements and have all their elements of order p were determined and characterized in terms independent of any particular selection of the generating elements. In dealing with fifty-seven of these groups there was no occasion to distinguish between one odd prime and another, except that in exhibiting canonical forms it was necessary to select irreducible polynomials and these, of course, depended on p. The fifty-eighth group was described in two ways in terms that were independent of p, but the proof of uniqueness could not be made without taking into account properties of p. These properties distribute the primes into classes, and the properties are reflected in the groups of order p11 in characteristic subgroups some of which exist for one prime and not for another. It may be that examination of the groups of isomorphisms of some of the fifty-seven groups would produce characteristic subgroups for one p that would not exist for another, but the writer considers it doubtful.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Brahana, H.R., Metabelian p∼grovps with five generators and orders p12 and pn, Illinois J. Math., 2 (1958), 641717.Google Scholar