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A Generalization of an Addition Theorem for Solvable Groups

Published online by Cambridge University Press:  20 November 2018

Thomas Yuster  and
Bruce Peterson
Thomas Yuster
Affiliation:
Middlebury College, Middlebury, Vermont
Bruce Peterson
Affiliation:
Middlebury College, Middlebury, Vermont
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The “sets” in this paper are actually multi-sets. That is, we allow an element to occur several times in a set and distinguish between the number of elements in a set and the number of distinct elements in the set. On the few occasions when we need to avoid repetition we will use the term “ordinary set.“

Definition. Let G be a group and let S a set of elements of G. An r-sum in S is an ordered subset of S of cardinality r; the result of that r-sum is the product of its elements in the designated order.

Definition. If S is a set, r(x, S) denotes the number of times x appears in S and [x, S] is a set consisting of r(x, S) copies of x. An n-set or n-subset is a set consisting of n elements. Hence [x, S] is an r(x, S)-subset of S.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 3 , 01 June 1984 , pp. 529 - 536
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Cauchy, A. L., Recherches sur les nombres, Journal Ecole Polytechnique 9 (1813), 99–123.Google Scholar
2. Erdös, , Ginsburg, and Ziv, , Theorem in additive number theory, Bull. Res. Coun. Israel 10 (1961), 41–43.Google Scholar