The sporadic groups

Abstract

AbstractIn this chapter we introduce the 26 sporadic simple groups. These are in many ways the most interesting of the finite simple groups, but are also the most difficult to construct. It is not possible here to provide complete proofs in all cases, but merely to indicate the general lines such proofs might take. Roughly speaking, proofs are given as far as the middle of Section 5.7, which deals with the Fischer groups. Section 5.2 deals with the large Mathieu groups M24, M23 and M22, and then the small Mathieu groups M12 and M11 are treated in Section 5.3. These groups have been known since Mathieu’s papers [130, 131, 132] of the 1860s and 1870s. In all these cases it is possible to give complete constructions in a reasonably small number of pages, computing the group orders and proving simplicity, as well as exhibiting a number of important subgroups. Other facts are stated with varying degrees of justification.KeywordsConjugacy ClassMaximal SubgroupOuter AutomorphismCongruence ClassSteiner SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.