1
Group G
Subgroup H
σ, τ ∈ G
Left coset σH
σH = { σ* ε | ε∈ H }
Left coset decomposition G = ∪j ∈ J τj H
Representative of left coset H τj
Corrected set of left cosets G/H = {τj H | j ∈ J }
Order | |
Index Number of G/H’ s elements ( G : H ) = | J |
Lagrange Theorem | G | = ( G : H ) | H |
From here, if | G | is prime number, G is cyclic group.
2
Group G
Subgroup N
N’ condition of normal subgroup
σ ∈G
σ N σ-1 = N
Center of G Cent ( G ) = {σ ∈ G | ∀τ ∈ G, σ* τ = τ*σ }
3
Group G
G’s normal subset N
Cosets’ set G/N
σ ∈G
Residue class of G/N ‘ condition
(σ1*N) (σ2*N) = (σ1*σ2) N
4
G1, G2 Group
Map φ : G1 → G2
σ, τ ∈ G
Group homomorphism for map φ φ(σ, τ ) = φ( σ ) * φ( τ )
Group isomorphism for map φ φ is bijective. ( φ ( a ) =φ (a’) ⇒ a = a’ and Image (φ) = G2 )
G1 and G2 are isomorphic. G1 ≅ G2
5
Homomorphism φ : G1 → G2
Unit of G1 e1
Unit of G2 e2
Kernel of homomorphism φ Ker (φ) = {σ ∈G | φ( σ ) = e2 }
Ker (φ) = {σ ∈G | φ( σ ) = e2 }
φ is injective Ker (φ) = { e1}
6
Homomorphism φ : G1 → G2
Homomorphism Theorem G1 / Ker (φ) ≅ Image (φ)
[Note]
Natural language probably has only left coset (or right coset), i.e., natural language is non-commutative. Chinese language <wo ai> means <I love>, but <ai wo> means <love me>.
[References]
Tokyo August 6, 2007
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