Group Theory  

In General > s.a. types of groups.
* History: The theory was invented by Galois in the early XIX century.
$ Def: A group is a pair (G, \(\circ\)), with G a set and \(\circ\) an operation \(\circ\) : G × GG, which is associative, has a neutral element, and such that each element has an inverse; I.e., a group is a monoid with inverse.
* Specifying a group: For finite groups, one can give the full multiplication table; In general, one gives a list of elements and a composition law in some representation, or a group presentation; The fundamental problem of group theory is the isomorphism problem, that of deciding whether any two finite presentations correspond to isomorphic groups; It was proved unsolvable in the late 1950s; > s.a. Combinatorial Group Theory; Group Presentation; Isomorphism; representations.
* Main branches: Abstract group theory; representation theory.
@ II: Falicov 66; Leech & Newman 69; Armstrong 88; Moody 94; Mirman 95 [II/III]; Snaith 03 [and rings, etc]; Ronan 06 [esp Monster group]; Reis 11.
@ III: Weyl 46; Kurosh 55; Hall 59; Hamermesh 62; Scott 64; Rotman 65; Suzuki 82; Zassenhaus 99; Roman 12; Machì 12 [many exercises].
@ Geometric group theory: de la Harpe 00; Chiswell in Bullett et al 17.
@ Books, special emphasis: Babakhanian 72 [cohomological methods]; Asche 89 [with software]; Geoghegan 08 [topological methods]; Steeb et al 12 [problems and solutions].
@ Related topics: Stillwell BAMS(82) [isomorphism problem].
> Applications: see algebraic topology; ordinary and partial differential equations; Group Theory in Physics below.
> Online resources: see Wikipedia page.

Related Concepts and Structure > s.a. Center; Coset; group action on a set; lie group [metric]; measure; Rank; Word.
* Conjugate elements: Two elements g1, g2 in G (two subgroups H1, H2 of G) are conjugate if there exists a g in G such that g1 = g g2 g–1 (H1 = g H2 g–1); This is an equivalence relation.
* Conjugacy Classes:
* Subgroups: Only if N is a normal subgroup is G/N a group; The order of a subgroup must be a divisor of the order of the group (think about cosets).
* Sylvester graph of a group: > see, e.g., matter.
@ References: Balantekin AIP(10)-a1011 [character expansions of products of invariant functions on G].

Operations on Groups > s.a. lie algebras [group contractions]; Semidirect Product.
* Extension of a group A by a group C: A new group BA with B/A = C; Expressed by the exact sequence

0 → ABC → 0 .

* Free product:

Subgroups > s.a. group actions [stabilizer]; Little Group; Torsion Subgroup.
* Normal: (H \(\triangleleft\) G) An H such that for all g in G, g–1Hg = H; Alternatively, H is the kernel of some homomorphism.
* Normalizer of a subset S: The subgroup of G defined by NG(S):= {gG | g–1Sg = S} ≡ {gG | Sg = gS}, i.e., the biggest subgroup of G in which S is normal; If S is a subgroup, S \(\triangleleft\) N(S), and S \(\triangleleft\) G is equivalent to N(S) = G; Special cases: If S = {s}, one element, N(S) = C(S), but in general C(S) ⊂ N(S).
* Commutant: The normal subgroup Q of elements of the form q = q1 q2 ... qm, where qi = gg'g–1g'–1; Remark: The quotient group G/Q is Abelian.
* Centralizer of a subset S: The subgroup of G defined by

CG(S):= {gG | g–1sg = s ,sS} ≡ {gG | gs = sg ,g ∈ S} .

Generalizations > s.a. Groupoid; loop; lie group; Monoid; Pseudogroup; quantum group; Semigroup.
* 2-group: A "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G × GG has been replaced by a functor.
@ General references: Barrett & Mackaay TAC(06)m.CT/04 [categorical groups, representations]; Davvaz 12 [polygroup theory].
@ 2-groups: Baez & Lauda TAC(04)m.QA/03; Baez et al MAMS(12)-a0812 [infinite-dimensional representations]

And Physics > s.a. gauge theory; general relativity; lie groups; quantum theory and canonical approach; symmetries.
* Idea: It is a very effective way of formulating and exploiting the symmetries of a system; The knowledge of a symmetry group gives the level scheme, branching ratios and selection rules; Symmetries are usually broken, but the approximation is still useful.
* History: Symmetry in physics was introduced with special relativity and the Lorentz group; Noether's theorem; 1932, Heisenberg joins p and n into an SU(2) doublet and postulates an SU(2)-invariant Lagrangian; 1954, local symmetries introduced with Yang-Mills theory.
@ Books and overviews: Kahan 65; Wybourne 74; Balachandran & Trahern 84; Cornwell 84; Tung 85 (and Aivazis 91); Isham 89; Sternberg 91; Fässler & Stifel 92; Ludwig & Falter 96; Teodorescu & Nicorovici 04; Ma & Gu 04 [problems and solutions]; Bonolis RNC(04) [history]; Ma 07; Dresselhaus et al 08 [in condensed matter, r PT(08)nov]; Ramond 10; Balachandran et al 10 [and Hopf algebras]; Jeevanjee 11 [r PT(12)apr]; Belhaj a1205-ln; Coddens 15 [spin and quantum mechanics]; Zee 16 [in a nutshell].
@ For particle physics: Barnes 10 [standard model]; Haywood 10 [symmetries and conservation laws]; Costa & Fogli 12; Saleem & Rafique 12.
@ And dynamics: Aldaya & Azcárraga FdP(87).

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