Let π(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G, denoted Γ G , is the graph with vertex set π(G) with edges {p, q} ∈ E(Γ G ) if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.

Abstract. In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper.

The objective of this paper is to develop criteria that guarantee that a finite group G which acts faithfully on a vector space V possesses "many" orbits of different sizes. This has consequences on (classical and modular) character degrees of finite solvable groups.Introduction.

No abstract

No abstract