A group, G, is a set with a multiplication rule defined on it which must satsfy:

- gi*gj in G forall gi*gj (closure)
- gi*(gj*gk)=(gi*gj)*gk (associativity)
- There exists identity, e, s.t. e*g=g=g*e
- forall g in G there exists g-1 (g inverse) s.t. g*g-1=g-1*g=e

Groups express symmetries. There are many examples of symmetries in nature and this is why groups are useful in physics.

Some theorems:

- H,K subgroups of G => H intersect K subgroup of G
- H,N subgroups of G with N normal => H intersect N subgroup of H is normal w.r.t H
- H,N normal subgroups of G => H intersect N subgroup of H is normal w.r.t G

source

jl@crush.caltech.edu index

Linear_Transformation_Group

translation_group

orthogonal_group

group_generation

continuous_group

symmetric_group

normal_subgroup

euclidean_group

tensor_product

representation

poincare_group

direct_product

covering_group

unitary_group

lorentz_group

group_algebra

compact_group

associativity

completeness

space_group

point_group

lie_algebra

equivalent

direct_sum

reducible

lie_group

isomorphi

conjugacy

character

subgroup

infinite

identity

dihedral

unitary

example

abelian

volume

tensor

simple

cyclic

center

order

coset

Tg

SU

SP

SO

SL

GL

Dn

Cn