Conjugacy classes of p-regular elements in p-solvable groups

Summary

We shall assume that any group is finite. One of the classic problems in Group Theory is to study how the structure of a group G determines properties on its conjugacy class sizes and reciprocally how these class sizes influence the structure of G. During the nineties several authors studied this relation by defining and studying two graphs associated to the conjugacy class sizes. In 1990 (see [8]), E. Bertram, M. Herzog and A. Mann defined a graph Γ (G) as follows: the vertices of Γ (G) are represented by the non-central conjugacy classes of G and two vertices C and D are connected by an edge if |C| and |D| have a common prime divisor. Later, this graph was studied in [12] and also used in [9] to obtain properties on the structure of G when some arithmetical conditions are imposed on the conjugacy class sizes. On the other hand, in 1995, S. Dolfi [14] studied a dual graph, Γ*(G), defined in the following way: the set of vertices are the primes dividing some conjugacy class size of G and two primes r and s are joined by an edge if rs divides some conjugacy class size of G. Independently, G. Alfandary also obtained some properties of these graphs (see [1]).

We shall suppose that G is a p-solvable group for some prime p. We shall consider the set of p-regular classes in G, that is, the conjugacy classes of p′-elements in G.