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Group properties characterised by configurations. (English) Zbl 1067.43001

This paper is motivated by a result of J. M. Rosenblatt and G. A. Willis [Can. Math. Bull. 44, 231–241 (2001; Zbl 0980.43001)] in which the notion of configuration for a finitely generated group, \(G\), was introduced and the amenability of \(G\) was characterized by its configurations. Starting from the fact that the amenability can be viewed as a finiteness condition, the authors of this paper investigate various finiteness properties of groups that can be characterized by configurations. The first such property is that of being periodic. They prove that if \(G\) is a finitely generated group having an element of infinite order, then there is a configuration set of \(G\) that is not a configuration set of any periodic group. In the remaining sections of the paper the authors examine the question of what it means for two groups to have the same configuration sets. They establish the following results:
If \(n\) is the minimum number of elements needed to generate a group \(H\) and if \(G\) is a finitely generated group configuration contained in \(H\) (this means that each configuration set for \(G\) is a configuration set for \(H \)), then at least \(n\) elements are required to generate \(G\). If \(G\) is a finite group and \(H\) has the same configuration sets as \(G \), then \(H\) is isomorphic to \(G\). If a finitely generated group \(G\) satisfies a semigroup law, and if \(H \) is configuration contained in \(G\), then \(H\) satisfies the same law. In particular, if \(G\) is abelian, then \(H\) is abelian, and if \(G\) is nilpotent of class \(c\), then \(H\) is nilpotent of class \(c\). If a finitely generated group \(H\) is configuration contained in the free group of rang \(n>0\), \(F_{n}\), then \(H\) is isomorphic to \(F_{n}\). If \(G\) and \(H\) are finitely generated abelian groups which have the same configuration sets, then \(G\) and \(H\) are isomorphic.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
20F05 Generators, relations, and presentations of groups
20F18 Nilpotent groups

Citations:

Zbl 0980.43001
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