Conway group

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co₁, Co₂ and Co₃ along with the related finite group Co₀ introduced by (Conway 1968, 1969).

The largest of the Conway groups, Co₀, is the group of automorphisms of the Leech lattice Λ with respect to addition and inner product. It has order

8,315,553,613,086,720,000

but it is not a simple group. The simple group Co₁ of order

4,157,776,806,543,360,000 =  2²¹ · 3⁹ · 5⁴ · 7² · 11 · 13 · 23

is defined as the quotient of Co₀ by its center, which consists of the scalar matrices ±1. The groups Co₂ of order

42,305,421,312,000 =  2¹⁸ · 3⁶ · 5³ · 7 · 11 · 23

and Co₃ of order

495,766,656,000 =  2¹⁰ · 3⁷ · 5³ · 7 · 11 · 23

consist of the automorphisms of Λ fixing a lattice vector of type 2 and type 3, respectively. As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co₁.

The inner product on the Leech lattice is defined as 1/8 the sum of the products of respective co-ordinates of the two multiplicand vectors; it is an integer. The square norm of a vector is its inner product with itself, always an even integer. It is common to speak of the type of a Leech lattice vector: half the square norm. Subgroups are often named in reference to the types of relevant fixed points. This lattice has no vectors of type 1.