We started our research with the intent on answering the following question: can we find a way to calculate all the subgroups of the symmetric group. This is easier said that done, as the number of subgroups for a symmetric group grows quickly with each successive symmetric group. This problem can actually be simplified to finding the subgroup conjugacy classes. So now we have the slightly diff...

For any countable group with infinite conjugacy classes we construct a family of forests on the group. each them there is random walk property that its sample paths almost surely converge to geometric boundary forest in way resembles simple walks trees. It allows us identify Poisson and show action free (which, particular, implies non-triviality boundary). As consequence obtain carries such sta...

in this paper, we will give necessary and sufficient conditions for certain hnnextensions of subgroup separable groups with normal associated subgroup to be conjugacyseparable. in fact, we will show that these hnn extensions are conjugacy separable if and onlyif the normalizer of one of its associated subgroup is conjugacy separable.

Abstract In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve problem in several classes groups. particular, start with free groups, that is, give an procedure find out if a finitely generated subgroup group is stable. Then further generalize result quasi-convex torsion-free hyperbolic groups limit

We apply Voronoi’s algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple Q-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.

We construct a hyperbolic group containing a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.

We apply Voronoi’s algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple Q -algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group.

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group.