May 19, 2009
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Design Theory:
Exquisite Geometries"By far the most important structure in design theory is the Steiner system
S(5, 8, 24). "-- "Block Designs," 1995, by Andries E. Brouwer
"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."
-- The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)
"... in 1861 Mathieu... discovered five multiply transitive permutation groups.... In a little-known 1931 paper of Carmichael... they were first observed to be automorphism groups of exquisite finite geometries."
The 1931 paper of Carmichael is now available online from the publisher for $10.
