# Article

 Title: A characterization of the linear groups $L_{2}(p)$ (English) Author: Khalili Asboei, Alireza Author: Iranmanesh, Ali Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 64 Issue: 2 Year: 2014 Pages: 459-464 Summary lang: English . Category: math . Summary: Let $G$ be a finite group and $\pi _{e}(G)$ be the set of element orders of $G$. Let $k \in \pi _{e}(G)$ and $m_{k}$ be the number of elements of order $k$ in $G$. Set ${\rm nse}(G):=\{m_{k}\colon k \in \pi _{e}(G)\}$. In fact ${\rm nse}(G)$ is the set of sizes of elements with the same order in $G$. In this paper, by ${\rm nse}(G)$ 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 ${\rm nse}(G)$ consists of $1$, $p^{2}-1$, $p(p+\epsilon )/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than $3$ with $p \equiv 1$ modulo $4$, then $G \cong L_{2}(p)$. (English) Keyword: element order Keyword: set of the numbers of elements of the same order Keyword: linear group MSC: 20D06 idZBL: Zbl 06391505 idMR: MR3277747 DOI: 10.1007/s10587-014-0112-y . Date available: 2014-11-10T09:47:10Z Last updated: 2016-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/144009 . Reference: [1] Asboei, A. K., Amiri, S. S. S., Iranmanesh, A., Tehranian, A.: A characterization of symmetric group $S_{r}$, where $r$ is prime number.Ann. Math. Inform. 40 (2012), 13-23. Zbl 1261.20025, MR 3005112 Reference: [2] Asboei, A. K., Amiri, S. S. S., Iranmanesh, A., Tehranian, A.: A new characterization of $A_{7}$ and $A_{8}$.in An. Ştiinţ. Univ. Ovidius'' Constanţa Ser. Mat 21 (2013),43-50. MR 3145090 Reference: [3] Asboei, A. K., Amiri, S. S. S., Iranmanesh, A., Tehranian, A.: A new characterization of sporadic simple groups by nse and order.J. Algebra Appl. 12 (2013), Paper No. 1250158. MR 3005607, 10.1142/S0219498812501587 Reference: [4] Brauer, R., Reynolds, W. F.: On a problem of E. Artin.Ann. Math. 68 (1958), 713-720. Zbl 0082.24803, MR 0100635, 10.2307/1970164 Reference: [5] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal subgroups and ordinary characters for simple groups.Clarendon Press Oxford (1985). Zbl 0568.20001, MR 0827219 Reference: [6] Frobenius, G.: Verallgemeinerung des Sylow'schen Satzes.Berl. Ber. (1895), German 981-993. Reference: [7] Khatami, M., Khosravi, B., Akhlaghi, Z.: A new characterization for some linear groups.Monatsh. Math. 163 (2011), 39-50. Zbl 1216.20022, MR 2787581, 10.1007/s00605-009-0168-1 Reference: [8] Mazurov, V. D., Khukhro, E. I., eds.: The Kourovka Notebook. Unsolved Problems in Group Theory. Including archive of solved problems.Institute of Mathematics, Russian Academy of Sciences, Siberian Div. Novosibirsk (2006). MR 2263886 Reference: [9] Shao, C., Jiang, Q.: A new characterization of Mathieu groups.Arch. Math., Brno 46 (2010), 13-23. Zbl 1227.20007, MR 2644451 Reference: [10] Shao, C., Shi, W., Jiang, Q.: Characterization of simple $K_{4}$-groups.Front. Math. China 3 (2008), 355-370. Zbl 1165.20020, MR 2425160, 10.1007/s11464-008-0025-x Reference: [11] Shen, R., Shao, C., Jiang, Q., Shi, W., Mazurov, V.: A new characterization of $A_{5}$.Monatsh. Math. 160 (2010), 337-341. Zbl 1196.20032, MR 2661315, 10.1007/s00605-008-0083-x Reference: [12] Shi, W.: A new characterization of the sporadic simple groups.Group Theory. Proceedings of the Singapore group theory conference 1987 K. N. Cheng et al. Walter de Gruyter Berlin (1989), 531-540. Zbl 0657.20017, MR 0981868 Reference: [13] Zhang, L., Liu, X.: Characterization of the projective general linear groups $PGL(2,q)$ by their orders and degree patterns.Int. J. Algebra Comput. 19 (2009), 873-889. Zbl 1189.20017, MR 2589419, 10.1142/S0218196709005433 .

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