A Characterization of Projective Special Unitary Group U3(7) by nse

Let a group and be the set of element orders of . Let and let be the number of elements of order in . Let nse . In Khatami et al. and Liu's works, and are uniquely determined by nse . In this paper, we prove that if is a group such that nse = nse , then . 1. Introduction A finite group is called a simple -group if is a simple group with . In 1987, J. G. Thompson posed a very interesting problem related to algebraic number fields as follows (see [1]). Thompson’s Problem. Let and , where is the number of elements with order . Suppose that . If is a finite solvable group, is it true that is also necessarily solvable? It is easy to see that if and are of the same order type, then The proof is as follows: let be a group and some simple -group, where ; then if and only if and nse (see [2, 3]). And also the group is characterizable by order and nse (see [4]). Recently, all sporadic simple groups are characterizable by nse and order (see [5]). Comparing the sizes of elements of same order but disregarding the actual orders of elements in of Thompson’s problem, whether it can characterize finite simple groups? Up to now, some groups especial for , where , can be characterized by only the set nse (see [6, 7]). The author has proved that the group is characterizable by nse (see [8]). In this paper, it is shown that the group also can be characterized by nse. Here, we introduce some notations which will be used. Let denote the product of integer by integer . If is an integer, then we denote by the set of all prime divisors of . Let be a group. The set of element orders of is denoted by . Let and let be the number of elements of order in . Let nse . Let denote the set of prime such that contains an element of order . denotes the projective special linear group of degree over finite fields of order . denotes the projective special unitary group of degree over finite fields of order . The other notations are standard (see [9]). 2. Some Lemmas Lemma 1 (see [10]). Let G be a finite group and m a positive integer dividing . If , then . Lemma 2 (see [11]). Let G be a finite group and let be odd. Suppose that P is a Sylow p-subgroup of G and with . If P is not cyclic and , then the number of elements of order n is always a multiple of . Lemma 3 (see [7]). Let G be a group containing more than two elements. If the maximal number s of elements of the same order in G is finite, then G is finite and . Lemma 4 (see [12, Theorem ]). Let G be a finite solvable group and , where , . Let and let be the number of Hall -subgroups of G. Then, satisfies the following conditions for all