On groups satisfying .
The paper reports the results of a search for pairs of groups of order that can be placed in the distance for the case when . The constructions that are used are of the general character and some of their properties are discussed as well.
Let be a finite group. The prime graph of is a graph whose vertex set is the set of prime divisors of and two distinct primes and are joined by an edge, whenever contains an element of order . The prime graph of is denoted by . It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if is a finite group such that , where , then has a unique nonabelian composition factor isomorphic to or .