Nielsen equivalence in a class of random groups

Nielsen Equivalence in Small Cancellation Groups

Let G be a group given by the presentation 〈a1, . . . , ak , b1, . . . bk | ai = ui(b̄), bi = vi(ā) for 1 ≤ i ≤ k〉, where k ≥ 2 and where the ui ∈ F (b1, . . . , bk) and wi ∈ F (a1, . . . , ak) are random words. Generically such a group is a small cancellation group and it is clear that (a1, . . . , ak) and (b1, . . . , bk) are generating n-tuples for G. We prove for generic choices of u1, . . ....

The graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations

A Fractal Non-Contracting Class of Automata Groups

A NOTE ON THE COMMUTING GRAPHS OF A CONJUGACY CLASS IN SYMMETRIC GROUPS

The commuting graph of a group is a graph with vertexes set of a subset of a group and two element are adjacent if they commute. The aim of this paper is to obtain the automorphism group of the commuting graph of a conjugacy class in the symmetric groups. The clique number, coloring number, independent number, and diameter of these graphs are also computed.