Geodesic
A new characterization of sporadic Higman--Sims and Held groups
Eurasian mathematical journal, Tome 5 (2014) no. 3, pp. 102-116.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\mathrm{nse}(G)=\{s_k|k\in\omega(G)\}$. The projective special linear groups $L_3(4)$ and $L_3(5)$ are uniquely determined by $\mathrm{nse}$. In this paper, we prove that if $G$ is a group such that $\mathrm{nse}(G)=\mathrm{nse}(M)$ where $M$ is a sporadic Higman–Sims or Held group, then $G\cong M$. 
@article{EMJ_2014_5_3_a6,
     author = {Y. Yang and S. Liu},
     title = {A new characterization of sporadic {Higman--Sims} and {Held} groups},
     journal = {Eurasian mathematical journal},
     pages = {102--116},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2014_5_3_a6/}
}
TY  - JOUR
AU  - Y. Yang
AU  - S. Liu
TI  - A new characterization of sporadic Higman--Sims and Held groups
JO  - Eurasian mathematical journal
PY  - 2014
SP  - 102
EP  - 116
VL  - 5
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2014_5_3_a6/
LA  - en
ID  - EMJ_2014_5_3_a6
ER  - 
%0 Journal Article
%A Y. Yang
%A S. Liu
%T A new characterization of sporadic Higman--Sims and Held groups
%J Eurasian mathematical journal
%D 2014
%P 102-116
%V 5
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2014_5_3_a6/
%G en
%F EMJ_2014_5_3_a6
Y. Yang; S. Liu. A new characterization of sporadic Higman--Sims and Held groups. Eurasian mathematical journal, Tome 5 (2014) no. 3, pp. 102-116. http://geodesic.mathdoc.fr/item/EMJ_2014_5_3_a6/

[1] A. K. Asboei, “A new characterization of $PGL(2,p)$”, J. Algebra Appl., 12:7 (2013), 1350040, 5 pp. | DOI

[2] A. K. Asboei, S. S. S. Amiri, A. Iranmanesh, A. Tehranian, “A characterization of sporadic simple groups by NSE and order”, J. Algebra Appl., 12:2 (2013), 1250158, 3 pp. | DOI

[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Oxford University Press, 1985, xxxiv+252 pp.

[4] G. Frobenius, “Verallgemeinerung des sylowschen satze”, Journal of Mathematical Research with Applications. Berliner Sitz., 1895, 981–993

[5] M. Herzog, “On finite simple groups of order divisible by three primes only”, J. Algebra, 10 (1968), 383–388 | DOI

[6] J. M. Hall, The theory of groups, The Macmillan Co., New York, 1959, 141 pp.

[7] A. Jafarzadeh, A. Iranmanesh, “On simple $K_n$-groups for $n= 5, 6$”, Groups St. Andrews (2005), v. 2, London Math. Soc. Lecture Note Ser., 340, Cambridge Univ. Press, 2007, 517–526

[8] M. Khatami, B. Khosravi, Z. Akhlaghi, “A new characterization for some linear groups”, Monatsh. Math., 163:1 (2011), 39–50 | DOI

[9] S. Liu, “A characterization of projective special linear group $L_3(5)$”, Ital. J. Pure Appl. Math., 32 (2014), 203–212

[10] S. Liu, “A characterization of $L_3(4)$”, Science Asia, 39:4 (2013), 436–439 | DOI

[11] S. Liu, “A characterization of projective special unitary group $U_3(5)$ by NSE”, Arab J. Math. Sci., 20:1 (2014), 133–140 | DOI

[12] S. Liu, R. Zhang, “A new characterization of $A_{12}$”, Math. Sci., 6:6 (2012), 30 | DOI

[13] G. A. Miller, “Addition to a theorem due to Frobenius”, Bull. Amer. Math. Soc., 11:1 (1904), 6–7 | DOI

[14] C. Shao, Q. Jiang, “A new characterization of some linear groups by NSE”, J. Algebra Appl., 13:2 (2014), 1–9 | DOI

[15] C. G. Shao, W. J. Shi, Q. H. Jiang, “Characterization of simple $K_4$-groups”, Front. Math. China, 3:3 (2008), 355–370 | DOI

[16] C. G. Shao, W. J. Shi, Q. H. Jiang, “A characterization of simple $K_3$-groups”, Adv. Math. (China), 38:3 (2009), 327–330

[17] R. Shen, C. Shao, Q. Jiang, W. Shi, V. Mazurov, “A new characterization of $A_5$”, Monatsh. Math., 160:3 (2010), 337–341 | DOI

[18] W. J. Shi, “A new characterization of the sporadic simple groups”, Group theory (Singapore, 1987), de Gruyter, 1989, 531–540

[19] W. J. Shi, “On simple $K_4$-group”, Chin. Sci. Bul., 36 (1991), 1281–1283 (in Chinese)

[20] Q. Zhang, W. Shi, “Characterization of $L_2(16)$ by $\tau_e(L_2(16))$”, J. Math. Res. Appl., 32:2 (2012), 248–252