scholarly journals Elliptic curves and Thompson's sporadic simple group

2021 ◽  
Vol 224 ◽  
pp. 274-306
Author(s):  
Maryam Khaqan
Keyword(s):  
Elliptic Curves ◽  
Simple Group ◽  
Sporadic Simple Group

scholarly journals On (2,3)-generation of Fischer's largest sporadic simple group Fi24′

2019 ◽  
Vol 357 (5) ◽  
pp. 401-412
Author(s):  
Faryad Ali ◽  
Mohammed Ali Faya Ibrahim ◽  
Andrew Woldar
Keyword(s):  
Simple Group ◽  
Sporadic Simple Group

NATURAL EXISTENCE PROOF FOR LYONS SIMPLE GROUP

2003 ◽  
Vol 02 (03) ◽  
pp. 277-315
Author(s):  
GERHARD O. MICHLER ◽  
MICHAEL WELLER ◽  
KATSUSHI WAKI
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Conjugacy Classes ◽  
Existence Proof ◽  
Sporadic Simple Group ◽  
Character Tables ◽  
The Given ◽  
Fold Cover

In this article we give a self-contained existence proof for Lyons' sporadic simple group G by application of the first author's algorithm [18] to the given centralizer H ≅ 2A11 of a 2-central involution of G. It also yields four matrix generators of G inside GL 111 (5) which are given in Appendix A. From the subgroup U ≅ (3 × 2A8) : 2 of H ≅ 2A11, we construct a subgroup E of G which is isomorphic to the 3-fold cover 3McL: 2 of the automorphism group of the McLaughlin group McL. Furthermore, the character tables of E ≅ 3McL : 2 and G are determined and representatives of their conjugacy classes are given as short words in their generating matrices.

On the cohomology of the sporadic simple group J4

1993 ◽  
Vol 113 (2) ◽  
pp. 253-266 ◽  
Author(s):  
David John Green
Keyword(s):  
Simple Group ◽  
Cohomology Ring ◽  
Sporadic Simple Group ◽  
Group Order ◽  
Integral Cohomology ◽  
Trivial Intersection ◽  
Extraspecial Group

In this paper we calculate part of the integral cohomology ring of the sporadic simple group J4; this group has order 221.33.5.7. 113.23.29.31.37.43. More precisely, we obtain all of the cohomology ring except for the 2-primary part. As the cohomology has already been written down [9] at the primes which divide the group order only once, we concentrate here on the primes 3 and 11. In both of these cases the Sylow p-subgroups are extraspecial of order p3 and exponent p. We use the method which identifies the p-primary cohomology with the ring of stable classes in the cohomology of a Sylow p-subgroup. The stable classes are all invariant under the action of the Sylow p-normalizer; and some time is spent finding invariant classes in the cohomology ring of , the extraspecial group. Section 2 studies the prime 11: the invariant classes are the stable classes, because the Sylow 11-subgroups have the Trivial Intersection (T.I.) property. In Section 3 we study the prime 3, and see that all conditions for invariant classes to be stable reduce to one condition.

scholarly journals Mathieu group M24 and modular forms

1985 ◽  
Vol 99 ◽  
pp. 147-157 ◽  
Author(s):  
Masao Koike
Keyword(s):  
Simple Group ◽  
Modular Forms ◽  
Cusp Forms ◽  
Sporadic Simple Group ◽  
Mathieu Group

In [6], Mason reported some connections between sporadic simple group M24 and certain cusp forms which appear in the ‘denominator’ of Thompson series assigned to Fisher-Griess’s group F1. In this paper, we discuss the ‘numerator’ of these Thompson series.

ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

2019 ◽  
Vol 102 (1) ◽  
pp. 77-90
Author(s):  
PABLO SPIGA
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Alternating Group ◽  
Permutation Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

scholarly journals A characterisation of almost simple groups with socle 2E6(2) or M(22)

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Chris Parker ◽  
M. Reza Salarian ◽  
Gernot Stroth
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Exceptional Group ◽  
Sporadic Simple Group ◽  
Group Of Lie Type ◽  
Almost Simple Groups

AbstractWe show that the sporadic simple group M(22), the exceptional group of Lie type

(3, 𝑞, 𝑟)-generations of Fischer's sporadic group Fi′24

2019 ◽  
Vol 22 (3) ◽  
pp. 453-489
Author(s):  
Faryad Ali ◽  
Mohammed Ali Faya Ibrahim ◽  
Andrew Woldar
Keyword(s):  
Simple Group ◽  
Present Article ◽  
Prime Divisors ◽  
Sporadic Simple Group ◽  
Sporadic Group

Abstract A group G is said to be {(l,m,n)} -generated if it can be generated by two suitable elements x and y such that {o(x)=l} , {o(y)=m} and {o(xy)=n} . In [J. Moori, {(p,q,r)} -generations for the Janko groups {J_{1}} and {J_{2}} , Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes {(p,q,r)} for which a finite non-abelian simple group is {(p,q,r)} -generated. In the present article, we partially answer this question for Fischer’s largest sporadic simple group {\mathrm{Fi}_{24}^{\prime}} by determining all {(3,q,r)} -generations, where q and r are prime divisors of {\lvert\mathrm{Fi}_{24}^{\prime}\rvert} with {3<q<r} .

scholarly journals Some subgroups of the Thompson group

1988 ◽  
Vol 44 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Robert A. Wilson
Keyword(s):  
Simple Group ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Non Local ◽  
Thompson Group

AbstractWe determine all conjugacy classes of maximal local subgroups of Thompson's sporadic simple group, and all maximal non-local subgroups except those with socle isomorphic to one of five particular small simple groups.

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