Group Properties Characterized by Two-sided Configurations

Abstract

In this paper, we define a new type of configurations as two-sided configurations, and investigate which group properties can be characterized by them. It is proved that for polycyclic torsion free groups, having the same finite quotient sets does not imply the (two-sided) configuration equivalence. We show that isomorphisms and configuration equivalences coincide for some free products of groups and a class of nilpotent groups.

Two-sided configuration equivalence and isomorphism

“…We proved that M = Nx for an x ∈ H. By normality of M, we conclude that Nx h = Nx, for every h ∈ H. Therefore q N (x) ∈ Z(H/N), where Z stands for the center of the group H/N. (b) If M contains the identity of H, then by the above Lemma, N is nothing but M. This means that M should be a subgroup of H. But there is no reason to say that M should contain e H , so it is not true to consider such an assumption, as it was in proofs of statements in the last section of [5].…”

A survey of configurations

“…To investigate the behavior of a group in some cases, we need to know how a special element acts from both left and right sides. In this section, we give a new type of configurations which depends on the left and right translations and we shall give a review of results from [13].…”

“…It is worth pointing out that two definitions of configurations are different (see [13]). In fact there is no direct correspondence between one-sided and two-sided configuration sets for the given generator and partition.…”

Configuration of Nilpotent Groups and Isomorphism

“…Remark 2.10. In [2] the authors gave a positive answer to the above question with another type of configuration, say two-sided configuration.…”

“…(1,2,3), c 2 = (2, 1, 5), c 3 = (3, 4, 1), c 4 = (4, 5, 5), c 5 = (4, 3, 5), c 6 = (5, 4, 2), c 7 = (5, 4, 4).…”