Linguistic Note 12 Coset / August 6, 2007

Linguistic Note

12

Coset

　　  TANAKA Akio

1

Group     G

Subgroup     H

σ, τ ∈ G

Left coset    σH

σH = { σ* ε | ε∈ H }

Left coset decomposition     G = ∪_j ∈ J τ_j H

Representative of left coset H    τ_j

Corrected set of left cosets    G/H = {τ_j 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.      G_1  ≅ 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]

 Notice for Model Simplified Level    Tokyo February 14, 2005

 Positioning of Quantum     Tokyo February 19, 2005

 Model Simplified Level     Tokyo February 5, 2005 

 Model Simplified Level 2     Tokyo February 12, 2005

Tokyo August 6, 2007

Sekinan Research Field of Language