Group properties characterised by configurations

Configuration of Nilpotent Groups and Isomorphism

“…We prove that if the group G has the same configuration set with the direct product of groups Z n × F, where F is a finite group, then G is isomorphic with Z n × F . This generalizes the result of Abdollahi, Rejali and Willis [1] saying that if G 1 is an abelian group with the same configuration sets as G 2 , then they are isomorphic.…”

“…In [1] the authors tried to generalize the latter result for nilpotent groups. It was shown that if G 1 is a finitely generated nilpotent group of class c having the same configuration sets with G 2 , then G 2 is a nilpotent group of class c. In this paper we show that if G 1 is a torsion free nilpotent group of Hirsch length h, then so is G 2 .…”

“…It was shown that amenability of G is characterized by its configuration equations. In [1], the authors investigated some properties of groups which can be characterized by configurations and studied the question of whether G is determined up to isomorphism by its configurations.…”

A constructive way to compute the Tarski number of a group

“…Rosenblatt and Willis introduced a concept for groups to show that for an infinite discrete amenable group or a non-discrete amenable group G a net of positive, normalized functions in L 1 (G) can be constructed such that this net converges weak* to invariance but does not converge strongly to invariance [8]. This concept which is called configuration is also used to classify some group theoretical properties (see for example [1,2]).…”

Configuration equivalence is not equivalent to isomorphism

“…We also interested in investigating the question: For which subclasses of the class G of all groups, does configuration equivalence coincide with isomorphism? In [1], this question was answered positively for the class of finite, free and Abelian groups. In [2], it was shown that those groups with the form of Z n ×F , where Z is the group of integers, n ∈ N and F is an arbitrary finite group, are determined up to isomorphism by their configurations.…”