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In the area of modern algebra known as group theory, the Higman–Sims group ''HS'' is a sporadic simple group of order : 293253711 : = 44352000 : ≈ 4. ==History== ''HS'' is one of the 26 sporadic groups and was found by . They were attending a presentation by Marshall Hall on the Hall–Janko group ''J''2. It happens that ''J''2 is a permutation group of 100 points, and the stabilizer of a point is a subgroup with two other orbits of lengths 36 and 63. Inspired by this they decided to check for other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M22, which has permutation representations on 22 and 77 points. (The latter representation arises because the M22 Steiner system has 77 blocks.) By putting together these two representations, they found ''HS'', with a one-point stabilizer isomorphic to M22. ''HS'' is 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 ''HS'' is a transitive group of permutations of a 100 element set. independently discovered the group as a doubly transitive permutation group acting on a certain 'geometry' on 176 points. The Schur multiplier has order 2, the outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Higman–Sims group」の詳細全文を読む スポンサード リンク
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