Finite Groups

In General [> s.a. groups.]
* Types: The simple ones include finite Chevalley, permutation (symmetric), alternating, and 26 sporadic groups.
* Idea: They are all made of groups which are for the most part imitations of Lie groups using finite fields (e.g., GL(n,q), the general n-dimensional linear group over a field of q elements)
@ References: Coxeter & Moser 72; Gorenstein BAMS(79); Huppert & Blackburn 82; Aschbacher BAMS(79), 86; Smith BAMS(97) [polynomial invariants]; Huppert 98 [character theory].

Finite Simple Groups
* Situation: The problem of classifying finite simple groups is over; Now the interest is in the connections with topology, representations, and sporadic groups.
* Monstrous Moonshine: In 1978, John McKay made the intriguing observation that 196884 = 196883 + 1; Monstrous Moonshine is the field inspired by this observation, which started with Conway and Norton's 1979 paper, proposing a completely unexpected relationship between finite simple groups and modular functions.
@ Classification: Gorenstein et al 94; Solomon BAMS(01) [history].
@ Related topics: Gannon m.QA/01-in, m.QA/04 [Monstrous Moonshine]; Kriz & Siegel SA(08)jul [sporadic simple groups].

Chevalley Groups
* Idea: The finite Chevalley groups arise when the parameters in a simple or reductive Lie group are replaced by elements of a finite field; They include most finite simple groups.
@ References: Srinivasan 79.

Permutation (or Symmetric) Group > s.a. partitions.
$ Def: The group S_n of all permutations of n objects.
$ Alternating group: The subgroup of the permutation group consisting of even permutations.
@ Representations: Chaturvedi et al PLA(08) [Schwinger representation].

Other Groups > s.a. Coxeter; Icosahedral.
* Examples: The 8-element group { 1, i, j, k}, or { 1, i¹, i², i³}.

Representations [> s.a. group representations.]
* Results: (i) Need to deal only with permutation groups; Use Young tableaux (over the rational field) or Specht modules; (ii) The number of inequivalent irr's is equal to the number of classes; (iii) A group of order n has r irreps, of order ₁, ₂, ..., _r, respectively, if ₁² + ₂² + ... + _r² = n.
* Example: S₃, the permutation group on 3 elements, of order 6, has 3 classes (the identity, the two cyclic elements, and the 3 pairwise interchanges), so it has 3 irreps, of order ₁ = ₂ = 1 and ₃ = 2 (1² + 1² + 2² = 6); They are ⁽¹⁾ = {1, 1, 1, 1, 1, 1}, ⁽²⁾ = {1, 1, 1, –1, –1, –1}, and ⁽³⁾ (which can be given in unitary form).
@ References: Curtis & Reiner 62; Feit 82; Nagao & Tsushima 89; Collins 90; in Fulton & Harris 91.

Related Concepts and Results > s.a. SU(2).
* Davenport’s constant: For a finite abelian group G, D(G) is the smallest integer d such that every sequence of d elements (repetition allowed) in G contains a non-empty zero-sum subsequence.
* Sylow subgroup: The 2-Sylow subgroup of a finite group G is a subgroup of order 2ⁿ (n = 0, 1, 2, ...) not properly contained in a larger one; Sylow's first theorem: If G is a group of order n, p is prime and m N, such that p^m|n, then G has a subgroup of order p^m.
@ References: Dolgachev BAMS(08) [reflection groups].

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