Combinatorics/Algebra
On (2,3)-generation of Fischer's largest sporadic simple group $Fi24′$
Comptes Rendus. Mathématique, Volume 357 (2019) no. 5, pp. 401-412.

A group G is said to be $(2,3)$-generated if it can be generated by an involution x and an element y of order three. For G a sporadic simple group, it was proved by the third author Woldar (1989) [26] that G is $(2,3)$-generated if and only if $G∉{M11,M22,M23,McL}$. In this paper, we investigate all possible $(2,3)$-generations of Fischer's largest sporadic simple group $Fi24′$ under the assumption that the product xy has prime order.

Un groupe G est dit $(2,3)$-engendré s'il peut être engendré par une involution x et un élément y d'ordre trois. Pour un groupe simple sporadique G, il a été montré par le troisième auteur Woldar (1989) [26] que G est $(2,3)$-engendré si et seulement si $G∉{M11,M22,M23,McL}$. Nous étudions ici toutes les $(2,3)$-générations du plus grand groupe simple sporadique de Fischer $Fi24′$, en supposant que le produit xy est d'ordre premier.

Accepted:
Published online:
DOI: 10.1016/j.crma.2019.05.004

Faryad Ali 1; Mohammed Ali Faya Ibrahim 2; Andrew Woldar 3

1 Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O.Box 90950, Riyadh 11623, Saudi Arabia
2 Department of Mathematics, Najran University, Najran, Saudi Arabia
3 Department of Mathematics and Statistics, Villanova University, Villanova, PA 19085, USA
@article{CRMATH_2019__357_5_401_0,
author = {Faryad Ali and Mohammed Ali Faya Ibrahim and Andrew Woldar},
title = {On (2,3)-generation of {Fischer's} largest sporadic simple group $F{i}_{24}^{\phantom{\rule{0.2em}{0ex}}\prime }$},
journal = {Comptes Rendus. Math\'ematique},
pages = {401--412},
publisher = {Elsevier},
volume = {357},
number = {5},
year = {2019},
doi = {10.1016/j.crma.2019.05.004},
language = {en},
}
TY  - JOUR
AU  - Mohammed Ali Faya Ibrahim
AU  - Andrew Woldar
TI  - On (2,3)-generation of Fischer's largest sporadic simple group $F{i}_{24}^{\phantom{\rule{0.2em}{0ex}}\prime }$
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 401
EP  - 412
VL  - 357
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2019.05.004
LA  - en
ID  - CRMATH_2019__357_5_401_0
ER  - 
%0 Journal Article
%A Mohammed Ali Faya Ibrahim
%A Andrew Woldar
%T On (2,3)-generation of Fischer's largest sporadic simple group $F{i}_{24}^{\phantom{\rule{0.2em}{0ex}}\prime }$
%J Comptes Rendus. Mathématique
%D 2019
%P 401-412
%V 357
%N 5
%I Elsevier
%R 10.1016/j.crma.2019.05.004
%G en
%F CRMATH_2019__357_5_401_0
Faryad Ali; Mohammed Ali Faya Ibrahim; Andrew Woldar. On (2,3)-generation of Fischer's largest sporadic simple group $F{i}_{24}^{\phantom{\rule{0.2em}{0ex}}\prime }$. Comptes Rendus. Mathématique, Volume 357 (2019) no. 5, pp. 401-412. doi : 10.1016/j.crma.2019.05.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.05.004/

[1] F. Ali On the ranks of $O′N$ and Ly, Discrete Appl. Math., Volume 155 (2007) no. 3, pp. 394-399

[2] F. Ali On $(2,3,t)$-generations for the Conway group $Co1$, AIP Conf. Proc., Volume 1557 (2013), pp. 46-49

[3] F. Ali On the ranks of $Fi22$, Quaest. Math., Volume 37 (2014) no. 4, pp. 591-600

[4] F. Ali $(2,3,t)$-generations for the Suzuki's sporadic simple group Suz, Appl. Math. Sci. (Ruse), Volume 8 (2014) no. 45–48, pp. 2375-2381

[5] F. Ali; M. Al-Kadhi; A. Aljouiee; M. Ibrahim 2-Generations of finite simple groups in $GAP$, IEEE Conf. Proc. CSCI, Volume 249 (2016), pp. 1339-1344 | DOI

[6] F. Ali; M.A.F. Ibrahim On the simple sporadic group He generated by the $(2,3,t)$ generators, Bull. Malays. Math. Sci. Soc., Volume 35 (2012) no. 3, pp. 745-753

[7] M.D.E. Conder Hurwitz groups: a brief survey, Bull. Amer. Math. Soc., Volume 23 (1990) no. 2, pp. 359-370

[8] M.D.E. Conder; J. Siran; T. Tucker The genera, reflexibility and simplicity of regular maps, J. Eur. Math. Soc., Volume 12 (2010) no. 2, pp. 343-364

[9] M.D.E. Conder; R.A. Wilson; A.J. Woldar The symmetric genus of sporadic groups, Proc. Amer. Math. Soc., Volume 116 (1992), pp. 653-663

[10] J.H. Conway; R.T. Curtis; S.P. Norton; R.A. Wilson Atlas of Finite Groups, Oxford University Press (Clarendon), Oxford, UK, 1985

[11] L. Di Martino; C. Tamburini 2-Generation of finite simple groups and some related topics (A. Barlotti et al., eds.), Generators ans Relations in Groups and Geometry, Kluwer Academic Publishers, New York, 1991, pp. 195-233

[12] L. Di Martino; N. Vavilov $(2,3)$-generation of $SL(n,q)$. I. Cases $n=5,6,7$, Commun. Algebra, Volume 22 (1994) no. 4, pp. 1321-1347

[13] B. Fischer Finite groups generated by 3-transpositions. I, Invent. Math., Volume 13 (1971) no. 3, pp. 232-246

[14] S. Ganief; J. Moori $(2,3,t)$-generations for the Janko group $J3$, Commun. Algebra, Volume 23 (1995) no. 12, pp. 4427-4437

[15] J.L. Gross; T.W. Tucker Topological Graph Theory, Dover, 2001

[16] I.M. Isaacs Character Theory of Finite Groups, Dover, New York, 1994

[17] G.A. Jones Beauville surfaces and groups: a survey (R. Connelly; A.I. Weiss; W. Whiteley, eds.), Rigidity and Symmetry, Fields Institute Communications, vol. 70, 2014, pp. 205-225

[18] M.W. Liebeck; A. Shalev The probability of generating a finite simple group, Geom. Dedic., Volume 56 (1995), pp. 103-113

[19] M.W. Liebeck; A. Shalev Classical groups, probabilistic methods and $(2,3)$-generation problem, Ann. of Math. (2), Volume 144 (1996), pp. 77-125

[20] M.W. Liebeck; A. Shalev Simple groups, probabilistic methods, and the conjecture of Kantor and Lubotzky, J. Algebra, Volume 184 (1996), pp. 31-57

[21] S.A. Linton; R.A. Wilson The maximal subgroups of the Fischer groups $Fi24$ and $Fi24′$, Proc. Lond. Math. Soc., Volume 63 (1991) no. 3, pp. 113-164

[22] A.M. Macbeath Generators of linear fractional groups, Proc. Symp. Pure Math., Volume 12 (1969), pp. 14-32

[23] G.A. Miller On the groups generated by two operators, Bull. Amer. Math. Soc., Volume 7 (1901), pp. 424-426

[24] J. Moori $(p,q,r)$-generations for the Janko groups $J1$ and $J2$, Nova J. Algebra Geom., Volume 2 (1993) no. 3, pp. 277-285

[25] J. Moori $(2,3,p)$-generations for the Fischer group $F22$, Commun. Algebra, Volume 22 (1994) no. 11, pp. 4597-4610

[26] A.J. Woldar On Hurwitz generation and genus actions of sporadic groups, Ill. J. Math., Volume 33 (1989) no. 3, pp. 416-437

[27] A.J. Woldar Representing $M11$, $M12$, $M22$ and $M23$ on surfaces of least genus, Commun. Algebra, Volume 18 (1990), pp. 15-86

Cited by Sources:

This work was funded by the National Plan for Science, Technology and Innovation (MARRIFAH) - King Abdulaziz City for Science and Technology - Kingdom of Saudi Arabia, award number (13-MAT264-08).