A CHARACTERIZATION OF U4(2) BY NSE

‎Let $G$ be a finite group and $\omega(G)$ be the set of element orders of $G$‎. ‎Let $k\in\omega(G)$ and $m_k$ be the number of elements of order $k$ in $G$‎. ‎Let $ nse(G)=\{m_k|k\in \omega(G)\}$‎. ‎The aim of this paper is to prove that‎, ‎if $G$ is a finite group such that nse($G$)=nse($U_4(2)$)‎, ‎then $G\cong U_4(2)$.

NSE characterization of some alternating groups

Let p ≥ 5 be a prime and n ∈ {p, p + 1, p + 2}. Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set nse (G) = {mk(G) : k ∈ πe(G)}. In this paper, we show that if nse (An) = nse (G), p ∈ π(G) and p2 ∤ |G|, then G ≅ An. As a consequence of our result, we show that if nse (An) = nse (G) and |G| = |An|, then G ≅ An.

A characterization of Sporadic Janko groups \(J_{2}\) and \(J_{3}\)

Let \(G\) be a group and \(\omega(G)\) be the set of element orders of \(G\). Let \(k\in \omega(G)\) and \(s_{k}\) be the number of elements of order \(k\) in \(G\). Let nse\((G)=\{s_{k}\big|k\in \omega(G)\}\). In Asboei's work, the author proved that \(J_{1}\) is unique determined by nse\((G)\). In this paper, we prove that if \(G\) is a group such that nse\((G)=\)nse(\(H\)), where \(H=J_{2}\) or \(J_{3}\), then \(G\cong H\).

A new characterization of A7 and A8

Let G be a finite group and πe(G) be the set of element orders of G. Let k ∈ πe (G)and mk be the number of elements of order k in G. Set nse(G):={mk|k ∈ πe (G)}. It is proved that An are uniquely determined by nse(An), where n ∈ {4,5,6}. In this paper, we prove that if G is a group such that nse(G)=nse(An) where n ∈ {7,8}, then G ≅ An.

A NEW CHARACTERIZATION OF SOME LINEAR GROUPS BY NSE

Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set nse (G) ≔ {mk(G) | k ∈ πe(G)}, we call nse (G) the set of numbers of elements with same order. In this paper, we give a new characterization of simple linear group L2(2a) by its order |L2(2a)| and the set nse (L2(2a)), where either 2a - 1 or 2a + 1 is a prime.

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

Let G a group and ω(G) be the set of element orders of G. Let k∈ω(G) and let sk be the number of elements of order k in G. Let nse(G)={sk∣k∈ω(G)}. In Khatami et al. and Liu's works, L3(2) and L3(4) are uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse(U3(7)), then G≅U3(7).

A NEW CHARACTERIZATION OF PGL(2, p)

Let G be a finite group and πe(G) be the set of element orders of G. Let k ∈ πe(G) and mk be the number of elements of order k in G. Set nse (G) ≔ {mk ∣ k ∈ πe(G)}. In this paper, it is proved if G is a group with the following properties, then G ≅ PGL (2, p). (1) p > 3 is prime divisor of ∣G∣ but p2 does not divide ∣G∣. (2) nse (G) = nse ( PGL (2, p)).