On the Projective Special Unitary groups $PSU_3(q)$ and the sum of element orders

Abstract

‎In this paper‎, ‎we prove that projective special unitary groups $PSU_3(q)$‎, ‎where $q=2^n$ and $\frac{q^2-q+1}{\gcd(3,q+1)}$ is a prime number, can be uniquely determined by the even-order components of the group and the set of orders of centralizers of $p_{m}-$order elements in $G$ where $p_{m}$ is the largest element in $\pi(G)$.‎

‎In the following‎, ‎we shows that, in ‏a ‎‎‎special case, these groups can be recognized by using the sum of the group elements $\psi(G)=\sum_{x\in G}o(x)$ where $o(x)$ denotes the order of $x\in G$‎.‎