7 research outputs found
A new characterization of symmetric group by NSE
summary:Let be a group and be the set of element orders of . Let and be the number of elements of order in . Let nse. Assume is a prime number and let be a group such that nse nse, where is the symmetric group of degree . In this paper we prove that , if divides the order of and does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components
Thompson's conjecture for the alternating group of degree and
summary:For a finite group denote by the set of conjugacy class sizes of . In 1980s, J. G. Thompson posed the following conjecture: If is a finite nonabelian simple group, is a finite group with trivial center and , then . We prove this conjecture for an infinite class of simple groups. Let be an odd prime. We show that every finite group with the property and is necessarily isomorphic to , where
Recognition of some families of finite simple groups by order and set of orders of vanishing elements
summary:Let be a finite group. An element is called a vanishing element if there exists an irreducible complex character of such that . Denote by the set of orders of vanishing elements of . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let be a finite group and a finite nonabelian simple group such that and . Then . We answer in affirmative this conjecture for , where and either , or is a prime number, and , where and either or is a prime number
On the composition factors of a group with the same prime graph as
summary: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