Monday 30 April 2018

Linguistic Note 12 Coset


12

Coset


    TANAKA Akio

1
Group     G
Subgroup     H
στ  G
Left coset    σH
σH = { σ* ε | ε H }
Left coset decomposition     G =  J τH
Representative of left coset H    τj
Corrected set of left cosets    G/H = {τH | 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, is cyclic group.

2
Group     G
Subgroup     N
N’ condition of normal subgroup
σ G
σ N σ-1 = N
Center of G     Cent ( ) = {σ  τ  Gστ = 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) = (σ12N

4
G1G2     Group
Map     φ G1   G2
στ  G
Group homomorphism for map φ     φ(στ ) = φσ ) * φτ )
Group isomorphism for map φ      φ is bijective. ( φ ( a ) =φ (a’) ⇒ a = a’    and   Image (φ) = G)
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 | φσ ) = e}
Ker (φ) = {σ G | φσ ) = e}
φ is injective       Ker (φ) = { e1}

6
Homomorphism      φ G1   G2
Homomorphism Theorem     G/ 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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