Finite Groups| 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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