scholarly journals John Horton Conway. 26 December 1937—11 April 2020

2021 ◽  
Author(s):  
Robert Turner Curtis
Keyword(s):  
Number Theory ◽  
Automorphism Group ◽  
Free Will ◽  
Elementary Particles ◽  
Simple Groups ◽  
Half Century ◽  
Number Systems ◽  
Biographical Memoir ◽  
Sporadic Simple Groups ◽  
Free Will Theorem

John Conway was without doubt one of the most celebrated British mathematicians of the last half century. He first gained international recognition in 1968 when he constructed the automorphism group of the then recently-discovered Leech lattice, and in so doing discovered three new sporadic simple groups. At around the same time he invented The Game of Life, which brought him to the attention of a much wider audience and led to a cult following of Lifers. He also combined the methods of Cantor and Dedekind for extending number systems to construct what Donald Knuth (ForMemRS 2003) called ‘surreal numbers’, the achievement of which Conway was probably most proud. Throughout his life he continued to make significant contributions to many branches of mathematics, including number theory, logic, algebra, combinatorics and geometry, and in his later years he teamed up with Simon Kochen to produce the Free Will theorem, which asserts that if humans have free will then, in a certain sense, so do elementary particles. In this biographical memoir I attempt to give some idea of the depth and breadth of Conway's contribution to mathematics.

scholarly journals Local Fusion Graphs and Sporadic Simple Groups

10.37236/4298 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
John Ballantyne ◽  
Peter Rowley
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Odd Order ◽  
Vertex Set ◽  
Sporadic Simple Groups

For a group $G$ with $G$-conjugacy class of involutions $X$, the local fusion graph $\mathcal{F}(G,X)$ has $X$ as its vertex set, with distinct vertices $x$ and $y$ joined by an edge if, and only if, the product $xy$ has odd order. Here we show that, with only three possible exceptions, for all pairs $(G,X)$ with $G$ a sporadic simple group or the automorphism group of a sporadic simple group, $\mathcal{F}(G,X)$ has diameter $2$.

scholarly journals The Coset lattices of E. S. Barnes and G. E. Wall

1990 ◽  
Vol 49 (3) ◽  
pp. 418-433
Author(s):  
Alexander J. Hahn
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Piecewise Linear ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Special Cases ◽  
Sporadic Simple Groups ◽  
Dimension 2 ◽  
Simple Chevalley Group ◽  
Subsequent Literature

AbstractJohn Conway's analysis in 1968 of the automorphism group of the Leech lattice and his discovery of three sporadic simple groups led to the immediate speculation that other Z-lattices might have interesting automorphism groups which give rise to (possibly new) finite simple groups. (The classification theorem for the finite simple groups has since told us that no new finite simple groups can arise in this or any other way.) For example in 1973, M. Broué and M. Enguehard constructed, in every dimension 2n, an even lattice (unimodular if n is odd) whose automorphism group is related to the simple Chevalley group of type Dn. This family of integral lattices received attention and acclaim in the subsequent literature. What escaped the attention of this literature, however, was the fact that these lattices had been discovered years earlier. Indeed in 1959, E. S. Barnes and G. E. Wall gave a uniform construction for a large class of positive definite Z-lattices in dimensions 2n which include those of Broué and Enguehard as special cases. The present article introduces an abstracted and generalized version of the construction of Barnes and Wall. In addition, there are some new observations about Barnes-Wall lattices. In particular, it is shown how to associate to each such lattice a continuous, piecewise linear graph in the plane from which all the important properties of the lattice, for example, its minimum, whether it is integral, unimodular, even, or perfect can be read off directly.

scholarly journals Involutions in the Automorphism Groups of Small Sporadic Simple Groups

Algebra ◽  
2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Chris Bates ◽  
Peter Rowley ◽  
Paul Taylor
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Sporadic Simple Groups

For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators.

scholarly journals ON A DESIGN FROM PRIMITIVE REPRESENTATIONS OF THE FINITE SIMPLE GROUPS

2019 ◽  
pp. 771
Author(s):  
Ali Reza Rahimipour
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Full Automorphism Group ◽  
Sporadic Simple Groups ◽  
Design Construction

In this paper we present a design construction from primitive permutation representations of a finite simple group G. The group G acts primitively onthe points and transitively on the blocks of the design. The construction has this property that with some conditions we can obtain t-design for t >=2. We examine our design for fourteen sporadic simple groups. As a result we found a 2-(176,5,4) design with full automorphism group M22.

scholarly journals Commuting involution graphs for sporadic simple groups

2007 ◽  
Vol 316 (2) ◽  
pp. 849-868 ◽  
Author(s):  
C. Bates ◽  
D. Bundy ◽  
S. Hart ◽  
P. Rowley
Keyword(s):  
Simple Groups ◽  
Sporadic Simple Groups

scholarly journals On the Structure of Integral Group Rings of Sporadic Groups

2000 ◽  
Vol 3 ◽  
pp. 274-306 ◽  
Author(s):  
Frauke M. Bleher ◽  
Wolfgang Kimmerle
Keyword(s):  
Automorphism Groups ◽  
Symmetric Groups ◽  
Computational Procedure ◽  
Isomorphism Problem ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Sporadic Groups ◽  
Integral Group Rings ◽  
Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

scholarly journals Pointed Hopf algebras over the sporadic simple groups

2011 ◽  
Vol 325 (1) ◽  
pp. 305-320 ◽  
Author(s):  
N. Andruskiewitsch ◽  
F. Fantino ◽  
M. Graña ◽  
L. Vendramin
Keyword(s):  
Hopf Algebras ◽  
Simple Groups ◽  
Pointed Hopf Algebras ◽  
Sporadic Simple Groups
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