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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