Finite Groups  

In General > s.a. group theory.
* 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)
@ General references: Coxeter & Moser 72; Gorenstein BAMS(79); Huppert & Blackburn 82; Aschbacher BAMS(79), 86; Smith BAMS(97) [polynomial invariants]; Huppert 98 [character theory]; Wehrfritz 99.
@ And physics: Kornyak LNCS(11)-a1106 [finite quantum dynamics and observables in terms of permutations]; García-Morales a1505 [digital calculus].

Finite Simple Groups
* Idea: They include the finite Chevalley groups, permutation (symmetric) groups, alternating groups, and 26 sporadic groups.
* Situation: The problem of classifying finite simple groups is over; Now the interest is in the connections with topology, representations, and sporadic groups.
* Sporadic groups: The 26 exceptional groups found in the classification of finite simple groups.
* Monster Group: The largest sporadic group, a group with more than 1053 elements constructed in 1982 by Robert Griess; > s.a. Wikipedia page.
* Monstrous Moonshine: In 1978, John McKay made the intriguing observation that 19,6884 (the first important coefficient of the j-function of number theory) equals 19,6883 + 1 (the first two special dimensions of the monster group); John Thompson then noticed that 21,493,760 = 1 + 196,883 + 21,296,876; Monstrous Moonshine is the field inspired by these observations, which started with Conway and Norton's 1979 paper, proposing a completely unexpected relationship between finite simple groups and modular functions; Richard Borcherds proved that the bridge between the two subjects is in string theory, and earned a Fields Medal for this work, leading to the new field of generalized Kac-Moody algebras.
* Umbral Moonshine Conjecture: Formulated in 2012, it proposes that in addition to monstrous moonshine, there are 23 other moonshines, mysterious correspondences between the dimensions of a symmetry group on the one hand, and the coefficients of a special function on the other; The new moonshines appear to be intertwined with K3 surfaces, some of the most central structures in string theory.
@ Classification: Gorenstein et al 94; Solomon BAMS(01) [history]; Ivanov 18 [Mathieu groups].
@ Sporadic groups: Kriz & Siegel SA(08)jul; Boya Sigma(11)-a1101; Boya JPA(13)-a1305 [intro for physicists]; > s.a. MathWorld page; Wikipedia page.
@ Moonshine: Gannon m.QA/01-conf, m.QA/04 [Monstrous Moonshine]; Cheng et al CNTP-a1204 [Umbral Moonshine]; Klarreich Quanta(15) [and strings]; Kachru a1605-proc [elementary introduction]; Anagiannis & Cheng a1807 [TASI lecture notes]; Tatitscheff a1902 [short intro].

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.
@ References: Blessenohl & Schocker 05 [non-commutative character theory]; Chaturvedi et al PLA(08) [Schwinger representation].

Other Groups > s.a. Coxeter Groups; Icosahedral Group.
* Examples: The 8-element group {± 1, ± i, ± j, ± k}, or {± 1, ± iσ1, ± iσ2, ± iσ3}.
@ Crystallographic groups: Szczepański 12; > s.a. Wikipedia page.

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 irreducible representations, of order \(\lambda_1^{~}\), \(\lambda_2^{~}\), ..., \(\lambda_r^{~}\), respectively, if \(\lambda_1^2\) + \(\lambda_2^2\) + ... + \(\lambda_r^2\) = n.
* Example: The group \(S_3\) of permutations on 3 elements, of order 6, has 3 classes (the identity, the two cyclic elements, and the 3 pairwise interchanges), so it has 3 irreducible representations, of order \(\lambda_1^{~}\) = \(\lambda_2^{~}\) = 1 and \(\lambda_3^{~}\) = 2 (with \(1^2 + 1^2 + 2^2 = 6\)); They are \(\Gamma^{(1)}\) = {1, 1, 1, 1, 1, 1}, \(\Gamma^{(2)} = \{1, 1, 1, -1, -1, -1\}\), and \(\Gamma^{(3)}\) (which can be given in unitary form).
@ General references: Curtis & Reiner 62; Feit 82; Nagao & Tsushima 89; Collins 90; in Fulton & Harris 91; Sengupta 12; Steinberg 12.
@ Related topics: Moore & Russell a1009 [approximate representations].

Related Concepts and Results > s.a. SU(2) group.
* 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 \(2^n\) (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 ∈ \(\mathbb N\), such that \(p^m|n\), then G has a subgroup of order pm.
@ References: Dolgachev BAMS(08) [reflection groups].

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