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.

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

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