On the Breadth of a Finite p-Group.
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 .
Let be a normal subgroup of a group . The structure of is given when the -conjugacy class sizes of is a set of a special kind. In fact, we give the structure of a normal subgroup under the assumption that the set of -conjugacy class sizes of is , where , and are distinct primes for , .