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 = Zn_1 Zn_2 ... Zn_k ,
where Zn_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].
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 
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.
Groups from Other Structures > s.a. group
action.
* And structured sets or
categories: Each set, possibly with extra
structure (e.g., a diff 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 ht/97 [representations
and geon statistics]; Giulini mp/06 [and
canonical quantum gravity]; Leininger & McReynolds T&A(07)
[separable subgroups]; > s.a. theta sectors.
@ Metaplectic group: de Gosson 01; > s.a. modified
quantum mechanics.
Other Types > s.a. lie group [including
formal]; Homeotopy, Poisson-Lie
Group.
* Perfect group: A group G such
that its Abelianization G /
[G, G]
= {e}.
* Simple group: One with no (proper nontrivial) invariant subgroup.
@ General references: Kaplansky 71 [locally compact]; Beadon 83 [discrete];
Majid ht/92-in
[braided, intro].
@ Algebraic groups: Humphreys 75; Springer 81; Hochschild 81.
@ Transformation groups: tom Dieck 87.
@ With an order relation: Glass 81.
> Other types: see finite
groups [including
Chevalley]; 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 
mx.
$ Def: We call M-group
a quadruple (G, , M,
), with
(G,)
a group, M a set, and : M × G → G,
(m,x)
mx, such that m(x y)
= mx my.
* Examples: A Z-group is the same as an Abelian group.
@ References: in Goldhaber & Ehrlich 70.
Specific Groups > s.a. G2; lie
group examples; lorentz; poincaré.
* Heisenberg group/algebra:
Abstractly, with generators x, y and z,
it is associated with the commutation relations [x, y] = z,
[x, z] = 0, [y, z] = 0 (or [a, a*]
= 1); Or, with generators x, p, and I,
[q, p] = i
I
, [q, q]
= 0 , [p, p] = 0 .
@ Heisenberg: Baskerville & Majid JMP(93)ht/92 [braided];
Costella qp/95 [[p, q] 
i];
> s.a. Commutation Relations.
@
Heisenberg, representations: Mnatsakanova et al LMP(03)mp/02 [holomorphic
functions]; Brodlie qp/04-PhD
[classical and quantum mechanics]; Derezinski mp/05.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
25 jun 2008