About the Book
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 37. Chapters: Baby Monster group, Classification of finite simple groups, Conway group, Conway group Co1, Conway group Co2, Conway group Co3, Fischer group, Fischer group Fi22, Fischer group Fi23, Fischer group Fi24, Hall-Janko group, Harada-Norton group, Held group, Higman-Sims group, II25,1, Janko group J1, Janko group J3, Janko group J4, Leech lattice, List of finite simple groups, Lyons group, Mathieu group, Mathieu groupoid, Mathieu group M11, Mathieu group M12, Mathieu group M22, Mathieu group M23, Mathieu group M24, McLaughlin group (mathematics), Miracle Octad Generator, O'Nan group, Rudvalis group, Sporadic group, Suzuki sporadic group, Thompson sporadic group, Umbral moonshine. Excerpt: In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. (In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group Bn(q) has the same order as Cn(q) for q odd, n > 2; and the groups A8 = A3(2) and A2(4) both have orders 20160.) Notation: n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d.f.g, where d is the order of the group of "diagonal automorphisms," f is the order of the (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). Simplicity: Simple for p a prime number. Order: p Schur multiplier: Trivial. Outer automorphism group: Cyclic of order p 1. Other names: Z/pZ Remarks: These are the only simple groups that are not perfect. Simplicity: Solvable for n 1. Schur multiplier: 2 for n = 5 or n > 7, 6 for n = 6 or 7; see Covering groups of the alternating and symmetric groups Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for n = 6, it has order 4 (elementary abelian). Other names: Altn. There is an unfortunate conflict with the notation for the (unrelated) groups An(q), and some authors use various different fonts for An to distinguish them. In particular, in this article we make the distinction by setting the alternating groups An in Roman font and the Lie-type groups An(q) in italic. Isomorphi
