# A characterization of the linear groups ${L}_{2}\left(p\right)$

Alireza Khalili Asboei; Ali Iranmanesh

Czechoslovak Mathematical Journal (2014)

- Volume: 64, Issue: 2, page 459-464
- ISSN: 0011-4642

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topKhalili Asboei, Alireza, and Iranmanesh, Ali. "A characterization of the linear groups $L_{2}(p)$." Czechoslovak Mathematical Journal 64.2 (2014): 459-464. <http://eudml.org/doc/262005>.

@article{KhaliliAsboei2014,

abstract = {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):=\lbrace m_\{k\}\colon k \in \pi _\{e\}(G)\rbrace $. 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 $G \cong L_\{2\}(p)$.},

author = {Khalili Asboei, Alireza, Iranmanesh, Ali},

journal = {Czechoslovak Mathematical Journal},

keywords = {element order; set of the numbers of elements of the same order; linear group; sets of element orders; sets of numbers of elements of equal orders; projective special linear groups; nse; characterizable groups; order of groups},

language = {eng},

number = {2},

pages = {459-464},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {A characterization of the linear groups $L_\{2\}(p)$},

url = {http://eudml.org/doc/262005},

volume = {64},

year = {2014},

}

TY - JOUR

AU - Khalili Asboei, Alireza

AU - Iranmanesh, Ali

TI - A characterization of the linear groups $L_{2}(p)$

JO - Czechoslovak Mathematical Journal

PY - 2014

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 64

IS - 2

SP - 459

EP - 464

AB - 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):=\lbrace m_{k}\colon k \in \pi _{e}(G)\rbrace $. 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 $G \cong L_{2}(p)$.

LA - eng

KW - element order; set of the numbers of elements of the same order; linear group; sets of element orders; sets of numbers of elements of equal orders; projective special linear groups; nse; characterizable groups; order of groups

UR - http://eudml.org/doc/262005

ER -

## References

top- 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. (2012) Zbl1261.20025MR3005112
- 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. (2013) MR3145090
- Asboei, A. K., Amiri, S. S. S., Iranmanesh, A., Tehranian, A., 10.1142/S0219498812501587, J. Algebra Appl. 12 (2013), Paper No. 1250158. (2013) MR3005607DOI10.1142/S0219498812501587
- Brauer, R., Reynolds, W. F., 10.2307/1970164, Ann. Math. 68 (1958), 713-720. (1958) Zbl0082.24803MR0100635DOI10.2307/1970164
- 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). (1985) Zbl0568.20001MR0827219
- Frobenius, G., Verallgemeinerung des Sylow'schen Satzes, Berl. Ber. (1895), German 981-993. (1895)
- Khatami, M., Khosravi, B., Akhlaghi, Z., 10.1007/s00605-009-0168-1, Monatsh. Math. 163 (2011), 39-50. (2011) Zbl1216.20022MR2787581DOI10.1007/s00605-009-0168-1
- 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). (2006) MR2263886
- Shao, C., Jiang, Q., A new characterization of Mathieu groups, Arch. Math., Brno 46 (2010), 13-23. (2010) Zbl1227.20007MR2644451
- Shao, C., Shi, W., Jiang, Q., 10.1007/s11464-008-0025-x, Front. Math. China 3 (2008), 355-370. (2008) Zbl1165.20020MR2425160DOI10.1007/s11464-008-0025-x
- Shen, R., Shao, C., Jiang, Q., Shi, W., Mazurov, V., 10.1007/s00605-008-0083-x, Monatsh. Math. 160 (2010), 337-341. (2010) Zbl1196.20032MR2661315DOI10.1007/s00605-008-0083-x
- 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. (1989) Zbl0657.20017MR0981868
- Zhang, L., Liu, X., 10.1142/S0218196709005433, Int. J. Algebra Comput. 19 (2009), 873-889. (2009) Zbl1189.20017MR2589419DOI10.1142/S0218196709005433

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