Concepts and Theorems Required for Dimensionless Analysis and to Build a Fiber Bundle Geometry
|a.||for any a, b∈G there is a unique c∈G|
ab = c
|b.||for any a, b, and c∈G|
a(bc) = (ab)c (associativity)
|c.||there exists e∈G, for which|
ea = a for any a∈G
|d.||for any a∈G exists a′∈G|
a′a = e
Furthermore if for any a, b∈G
ab = ba, then the group is commutative (Abelian).
Definition: We call G1 group an invariant group under transformation s∈G
if for any a∈G1
G⊇G1 and sa = as.
In other words, s is a commutator of G.
Definition: We call S subgroup an invariant subgroup of group G if for
any a∈G and any s∈S
sa = as.
An invariant group is sometimes called a normal subgroup.
Definition: If N is a normal (invariant) subgroup of Gi , then aN a∈G
classes are compatible classes. That is, if
a ≠ b a,b≠G, then aN ⋂ bN = 0.