Symmetric Generation of Groups: With Applications to Many of the Sporadic Finite Simple GroupsSome of the most beautiful mathematical objects found in the last forty years are the sporadic simple groups. However, gaining familiarity with these groups presents problems for two reasons. First, they were discovered in many different ways, so to understand their constructions in depth one needs to study lots of different techniques. Second, since each of them is in a sense recording some exceptional symmetry in spaces of certain dimensions, they are by their nature highly complicated objects with a rich underlying combinatorial structure. Motivated by initial results which showed that the Mathieu groups can be generated by highly symmetrical sets of elements, which themselves have a natural geometric definition, the author develops from scratch the notion of symmetric generation. He exploits this technique by using it to define and construct many of the sporadic simple groups including all the Janko groups and the Higman-Sims group. This volume is suitable for researchers and postgraduates. |
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action acts transitively additional relation alternating group bifid maps Cayley diagram Cayley graph CDCR central commutes conjugacy class conjugation constructed contains control subgroup copy corresponding cycle shape cyclic group defined denote dodecad double coset enumerator edge element of order elementary abelian example factor Fano plane Figure finite fixed given group G group of order hexads homomorphic image interchanges intersection array involutions J₁ joined Klein map labelled Lemma letters linear group MAGMA Mathieu group Mathieu group M24 matrix maximal subgroup modulo monomial representation non-trivial normal subgroup notation Note obtain octad outer automorphism pairs partitions permutation group points progenitor projective line s₁ sextet Shape of Shape single cosets Steiner system subgroup isomorphic subsets subspace symmetric group symmetric presentation synthemes t₁ Table tetrads Theorem transpositions trio trivial u₁ unique vectors vertices xy)² πω