A characterization of the linear groups L
2(p) |
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Authors: | Alireza Khalili Asboei Ali Iranmanesh |
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Institution: | 1. Department of Mathematics, Farhangian University, Shariati Mazandaran, Babol, Iran 2. Department of Mathematics, College of Engineering, Buin Zahra Branch, Islamic Azad University, Buin Zahra, Iran 3. Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O.Box: 14115-137, Tehran, Iran
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Abstract: | Let G be a finite group and π e (G) be the set of element orders of G. Let k ∈ π e (G) and m k be the number of elements of order k in G. Set nse(G):= {m k : k ∈ π e (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L 2(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L 2(p)| and nse(G) consists of 1, p 2 ? 1, p(p + ?)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p ≡ 1 modulo 4, then G ? L 2(p). |
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