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* × *G* → *G*,
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
*g*_{1}, *g*_{2}
in *G* (two subgroups *H*_{1},
*H*_{2} of *G*) are conjugate
if there exists a *g* in *G* such that *g*_{1}
= *g* *g*_{2} *g*^{−1}
(*H*_{1} = *g* *H*_{2}
*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

0 → *A* → *B* → *C* → 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*^{−1}*Hg* = *H*;
Alternatively, *H* is the kernel of some homomorphism.

* __Normalizer of a subset S__:
The subgroup of

*

*

*C*_{G}(*S*):=
{*g* ∈ *G* | g^{−1}*sg* = *s
,* ∀*s* ∈ *S*} ≡ {*g* ∈ *G* | *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* × *G* → *G* 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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