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];
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,
± 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 \(\lambda_1^{~}\), \(\lambda_2^{~}\), ..., \(\lambda_r^{~}\), 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
*p*^{m}.

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

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