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 10^{53} 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].

@ __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].

**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, ± i*σ*^{1}, ± i*σ*^{2}, ± i*σ*^{3}}.

@ __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 *λ*_{1}, *λ*_{}_{2},
..., *λ*_{}_{r},
respectively, if *λ*_{}_{1}^{2}
+ *λ*_{}_{2}^{2} +
... + *λ*_{}_{r}^{2}
= *n*.

* __Example__: S_{3},
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 irreducible representations, of order *λ*_{}_{1}
= *λ*_{}_{2} =
1 and *λ*_{}_{3}
= 2 (1^{2 }+ 1^{2} +
2^{2} =
6); They are Γ^{(1)}
= {1, 1, 1, 1, 1, 1}, Γ^{(2)}
= {1, 1, 1, –1, –1, –1}, and Γ^{(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 *p*^{m}.

@ __References__: Dolgachev BAMS(08) [reflection groups].

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