Types and Examples of Groups  

Abelian Groups
$ Def: An Abelian group is a commutative group, a G such that for all g, hG, gh = hg.
* Finitely generated: A finitely generated Abelian group A can be written as AGT, 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 ni.
* Torsion subgroup: The subgroup T of finite order elements of a group G, T:= {gG | ∃ 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: SG, we say that (G, f) is a free group on S if, for any group H and map g: SH, there is a unique homomorphism m: GH, 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 mM 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 × GG, (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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