Group
Theory| Group
Theory |
In General > s.a. Combinatorial
Group Theory; Isomorphism; representations and types
of groups.
* History: The theory was invented by Galois in the early XIX century.
$ Def: A group is a
pair (G, 
),
with G a set and an
operation : G × G → G,
which is associative, has a neutral element, and such that each element has
an inverse; I.e., a group is a monoid with inverse.
* Specifying a group:
For finite groups, one can give the full multiplication table; In
general, one gives a list of elements and a composition law in some representation
(> s.a. combinatorial group theory), or a group
presentation.
* Fundamental problem of
group theory: The isomorphism problem.
* Main branches: Abstract
group theory; representation theory.
* Applications: In topology, > see algebraic
topology;
In physics, > see below
and symmetries.
@ II: Falicov 66; Leech & Newman 69; Armstrong 88; Moody 94; Mirman
95 [II/III]; Ronan 06 [esp Monster group]; Reis 11.
@ III: Weyl 46; Kurosh 55; Hall 59; Hamermesh 62; Scott 64; Rotman 65;
Suzuki 82; Zassenhaus 99; Roman 12; Machì 12 [many exercises].
@ Special: Babakhanian 72 [cohomological methods]; Asche 89 [with software]; de la Harpe 00 [geometric group theory].
Related Concepts and Structure > s.a. Center; Coset; group
action on a set; lie group [metric]; measure; Rank; Word.
* Conjugate elements:
Two elements g1, g2 in G (two
subgroups H1,
H2 of G) are conjugate
if there exists a g in G such that g1 = g g2 g–1 (H1 = g H2 g–1);
This is
an equivalence relation.
* Conjugacy Classes:
* Subgroups: Only if N is a normal subgroup is G/N a group;
The order of a subgroup must be a divisor of the order of the group (think
about cosets).
* Sylvester graph of a group: > see,
e.g., matter.
@ References: Balantekin AIP(10)-a1011 [character expansions of products of invariant functions on G].
Operations on Groups > s.a. lie
algebras [group contractions]; Semidirect Product.
* Extension of a group A by a group C: A new group B ⊃ A with B/A = C;
Expressed by the exact sequence
0 → A → B → C → 0 .
* Free product:
Subgroups > s.a. group actions [stabilizer]; Little
Group; Torsion Subgroup.
* Normal: (H 
G)
An H such that for all g in
G, g–1Hg = H;
Alternatively, H is the kernel of some homomorphism.
* Normalizer of a subset S:
The subgroup of G defined by NG(S):=
{g ∈ G | g–1Sg = S} ≡ {g ∈ G | Sg = gS},
i.e., the biggest subgroup of G in
which S is normal; If S is a subgroup, S N(S),
and S G is
equivalent to N(S) = G;
Special cases: If S = {s}, one element, N(S) = C(S),
but in general C(S) ⊂ N(S).
* Commutant: The normal
subgroup Q of elements of the form q =
q1 q2 ... qm,
where qi = gg'g–1g'–1;
Remark: The quotient group G/Q is Abelian.
* Centralizer of a subset S: The subgroup of G defined by
CG(S):= {g ∈ G | g–1sg = s , ∀s ∈ S} ≡ {g ∈ G | gs = sg , ∀g ∈ S} .
Generalizations > s.a. Groupoid;
loop; lie group; Monoid; Pseudogroup;
quantum group; Semigroup.
* 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].
@ 2-groups:
Baez & Lauda TAC(04)m.QA/03; Baez et al a0812 [infinite-dimensional representations]
And Physics > s.a. gauge theory;
general relativity; lie
groups; quantum
theory and canonical approach; symmetries.
* Idea: It is a very effective way of formulating and exploiting the
symmetries of a system; The knowledge of a symmetry group gives the level scheme,
branching ratios and selection rules; Symmetries are usually broken,
but the approximation is still useful.
* History: Symmetry in
physics was introduced with special relativity and the Lorentz group; Noether's
theorem; 1932, Heisenberg joins p and n into an
SU(2) doublet and postulates an SU(2)-invariant Lagrangian; 1954, local
symmetries introduced with Yang-Mills theory.
@ Books and overviews: Kahan 65; Wybourne 74; Balachandran & Trahern
84; Cornwell 84;
Tung 85 (and
Aivazis 91);
Isham 89; Sternberg 91; Fässler & Stifel 92; Ludwig & Falter
96; Teodorescu & Nicorovici 04; Ma & Gu 04 [problems
and solutions]; Bonolis RNC(04)
[history]; Ma 07;
Dresselhaus et al 08 [in condensed matter, r PT(08)nov];
Ramond 10;
Barnes 10 [standard model]; Balachandran et al 10 [and Hopf algebras]; Haywood 10 [symmetries and conservation laws]; Costa & Fogli 12; Jeevanjee 11 [r PT(12)apr].
@ And dynamics: Aldaya & Azcárraga FdP(87).
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send feedback and suggestions to bombelli at olemiss.edu – modified 11 apr 2012