Conway group

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

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

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

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. It is common to speak of the type of a Leech lattice vector: half the square norm. This lattice has no vectors of type 1. The groups Co₂ (of order 42,305,421,312,000) and Co₃ (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co₁.