A new characterization of alternating groups

Abstract

Let \(G\) be a finite group and let \(\pi_{e}(G)\) be the set of element orders of \(G \). Let \(k \in \pi_{e}(G)\) and let \(m_{k}\) be the number of elements of order \(k \) in \(G\). Set nse(\(G\)):=\(\{ m_{k} | k \in \pi_{e}(G)\}\). In this paper, we show that if \(n = r\), \(r +1 \), \(r + 2\), \(r + 3\) \(r+4\), or \(r + 5\) where \(r\geq5\) is the greatest prime not exceeding \(n\), then \(A_{n}\) characterizable by nse and order.