In this paper we introduce the graph Γsc(G) associated with a group G, called solvable conjugacy class (abbreviated as SCC-graph), whose vertices are nontrivial classes of G and two distinct C,D adjacent if there exist x∈C y∈D such that 〈x,y〉 is solvable. We discuss connectivity, girth, clique number, several other properties SCC-graph. One our results asserts only finitely many finite groups S...

In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show t...

Of course, in that problem we have to take into account that the class sizes impose restrictions on the group structure. E.g. if the sizes are {1, p}, then the nilpotency class has to be 2. More precisely: the class sizes of a p-group G are {1, p} iff |G′| = p (Knoche; see also Theorem 3 below). But we can ask, e.g., if, given any set S ≠ {1, p} of p-powers, does there exist a group of class 3 ...