Types and Examples of Groups |

**Abelian Groups**

$ __Def__: An Abelian group is
a commutative group, a *G* such that for all *g*, *h*
∈ *G*, *gh* = *hg*.

* __Finitely generated__:
A finitely generated Abelian group *A* can be written as *A*
≅ *G* ⊕ *T*, where *G* is a free Abelian group,
and *T* the torsion subgroup, of the form T = \({\mathbb Z}_{n_1}\)
⊕ \({\mathbb Z}_{n_2}\) ⊕ ... ⊕ \({\mathbb Z}_{n_k}\),
where each \({\mathbb Z}_{n_i}\) is cyclic of order
*n*_{i}.

* __Torsion subgroup__: The
subgroup *T* of finite order elements of a group *G*,
*T*:= {*g* ∈ *G* | ∃ *n* > 0
such that *ng* = 0}; > s.a. tilings.

@ __References__: Kaplansky 54 [infinite];
Fuchs 60;
Fuchs 70,
Griffith 70 [infinite];
Fuchs 15 [emph. homological algebra and set theory].

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Free Groups**

* __Idea__: Think of a group as
defined not by its composition table, but by a set of generators *S* and
a set of defining relations *D*, a presentation; One can ask if, given
any group *G*, there exists another group with no defining conditions,
i.e., *free*, to which *G* is homomorphic; The answer is yes.

$ __Def__: Given a set *S*,
a group *G* and a function *f*: *S* → *G*, we say
that (*G*, *f*) is a free group on *S* if, for any group
*H* and map *g*: *S* → *H*, there is a unique
homomorphism *m*: *G* → *H*, such that *g*
= *m* \(\circ\) *f*.

* __Result__: One can show that
*f*(*S*) is a set of generators of *G*, and that, for any
*S*, there is a unique (up to isomorphisms) free group on *S*.

* __Result__: If *G* is a
group with generator set *S*, then *G* is a homomorphic image of
some free group on *S*.

@ __References__: in Goldhaber & Ehrlich 70;
in Hilton & Stammbach 71.

> __Online resources__:
see Wikipedia page.

**Groups from Other Structures** > s.a. group action.

* __And structured sets or
categories__: Each set, possibly with extra structure (e.g., a differentiable
manifold) *X* defines the group of automorphisms of *X*; Each
category *A* defines the group of homomorphisms of *A*.

* __Mapping class group__:
The group Map(*M*) of equivalence classes of large diffeomorphisms
of a manifold; Consists (at least for 2D manifolds with punctures), of a pure
mapping class group + a braid group; Its inequivalent unitary irreducible
representations for a spatial manifold give rise to "theta sectors''
in theories of quantum gravity with fixed spatial topology.

* __Metaplectic group__: The group
of linear canonical transformations.

@ __Mapping class group__: Goldman AM(97) [action on moduli space of bundles];
Sorkin & Surya AIP(97)ht [representations and geon statistics];
Giulini in(07)mp/06 [and canonical quantum gravity];
Leininger & McReynolds T&A(07) [separable subgroups];
Andersen & Fjelstad LMP(10) [reducibility of quantum representations];
> s.a. theta sectors.

@ __Metaplectic group__:
de Gosson 97;
de Gosson 17;
> s.a. modified quantum mechanics.

**Other Types**
> s.a. finite groups [including Chevalley]; lie groups
[including formal]; Homeotopy Group; Poisson-Lie Group.

* __Perfect group__: A group *G*
such that its Abelianization *G* / [*G*, *G*] = {*e*}.

* __Simple group__: One with no (proper,
non-trivial) invariant subgroup.

@ __General references__: Kaplansky 71 [locally compact];
Majid ht/92-proc [braided, intro].

@ __Discrete groups__: Beardon 83;
Farenick et al a1209-CMP(14) [operator systems].

@ __Infinite-dimensional groups__:
Khesin & Wendt 09 [geometry];
Albeverio et al a1511-in
[groups of smooth paths with values in a compact Lie group, reps].

@ __Algebraic groups__: Humphreys 75;
Springer 81;
Hochschild 81.

@ __Transformation groups__: tom Dieck 87.

@ __With an order relation__: Glass 81,
99.

> __Other types__: see Coxeter,
Semisimple, Solvable,
Topological Group [including amenable].

**Groups with Operators**

* __Idea__: A generalization of the
notion of a group with the set of its endomorphisms; To each *m* ∈
*M* there corresponds an endomorphism *x* \(\mapsto\) *mx*.

$ __Def__: We call *M*-group a quadruple
(*G*, \(\circ\), *M*, *μ*), with (*G*, \(\circ\) ) a group,
*M* a set, and *μ*: *M* × *G* → *G*,
(*m*, *x*) \(\mapsto\) *mx*, such that *m*(*x*
\(\circ\) *y*) = *mx* \(\circ\) *my*.

* __Examples__: A \(\mathbb Z\)-group
is the same as an Abelian group.

@ __References__: in Goldhaber & Ehrlich 70.

**Other Generalizations** > s.a. loop;
lie group; quantum group.

* __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].

> __Other generalizations__: see Groupoid;
locality in quantum field theory [group with causality]; Monoid;
Pseudogroup; Semigroup.

**Specific Groups**
> s.a. G2; Heisenberg Group;
lie group examples; lorentz group;
poincaré group.

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send feedback and suggestions to bombelli at olemiss.edu – modified 19 jun 2019