Types and Examples of Groups| 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
ni.
* 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