Higman-Sims group - Definition and Overview

In mathematics, the Higman-Sims group is a finite sporadic simple group of order 44352000. It can be characterized as the simple subgroup of index two in the group of automorphisms of the Higman-Sims graph. The Higman-Sims graph has 100 nodes, so the Higman-Sims group, or HS, has a permutation representation of degree 100. The Conway groups Co₂ and Co_{3 also contain HS.}

HS can also be defined in terms of generators a and b and the following relations:

<math>a^2 = b^5 = (ab)^{11} = (ab^2)^{10} = [a, b]^5 = [a, b^2]^6 =

[a, bab]^3 = <math>

<math>ababab^2ab^{-1}ab^{-2}ab^{-1}ab^2abab(ab^{-2})^4 =<math>

<math>ab(ab^2(ab^{-2})^2)^2ab^2abab^2(ab^{-1}ab^2)^2 =<math>

<math>abab(ab^2)^2ab(ab^{-1})^2ab(ab^2)^2ababab^{-2}ab^{-1}ab^{-2} = 1.<math>

HS is named for Donald G. Higman and Charles C. Sims.