Quotient Groups

1 Cosets

Let G be a group with subgroup H. ...Consider the relation ...x ∼ y ...⇔ ...y = h x , some h ∊ H. ...[ equivalently ...y x^–1 ∊ H. ]
This is an equivalence relation: ...x = 1 x ; ...x = h^–1 y ; ...z = g y ⇒ z = g ( h x ) = ( g h ) x .

For any x ∊ G , the right coset of x in H , ......H x ...:= ...[ x ] ...= ...{ y ∊ G | y = h x , some h ∊ H } , ...i.e. the equivalence class of x in the above relation.
Analagously, the left coset of x in H , ......x H ...:= ...{ z ∊ G | z = x h , some h ∊ H } ...( using the relation ...x ∼ z ...⇔ ...z = x h , some h ∊ H. )

In general ...x H ...≠ ...H x .

The set of right cosets in H is denoted ...H G ...:= ...{ H x | x ∊ G } , ...and the set of left cosets in H is denoted ...G ⁄ H ...:= ...{ x H | x ∊ G }

2 Index Of A Subgroup

If G is finite, then the number of elements in a coset is the same as the number of elements in H , ...i.e. ...| H | .
For suppose H = { h₁ , ⋯ , h_n } , ...n distinct h_i, ...then ...H x = { h₁ x , ⋯ , h_n x } . ...
Then for i ≠ j , ...h_i x = h_j x ...⇒ ...h_i x x^–1 = h_j x x^–1 ...⇒ ...h_i = h_j ...contradiction.

So the number of cosets in G is | G | ⁄ | H | . ...This number is known as the index of H in G , ...and is denoted ...| G : H | .

 

3 Quotient Group

If G is an abelian group ( with group operation '+' ) then ...x ∼ y ...if ...x – y ∊ H , ...and the left and right cosets coincide, ...
i.e. ...x + H = H + x , ...∀ x . ...H G ...= ...G ⁄ H ...= ...{ x + H | x ∊ G}

Define the operation ...⊕ : ( G ⁄ H ) × ( G ⁄ H ) → ( G ⁄ H ) ...by ...( x + H ) ⊕ ( y + H ) ...= ...( x + y ) + H .

With this operation ...G ⁄ H ...is a group:

• associative: ...( ( x + H ) ⊕ ( y + H ) ) ⊕ ( z + H ) ...= ...( ( x + y ) + H ) ⊕ ( z + H )
  .........= ...( ( x + y + z) + H ) ...= ...( x + H ) ⊕ ( ( y + z ) + H ) ...= ...( x + H ) ⊕ ( ( y + H ) ⊕ ( z + H ) ) .
• identity: ...( 0 + H ) ⊕ ( x + H ) ...= ...( 0 + x ) + H ...= ...x + H
• inverse: ...( x + H ) ⊕ ( –x + H ) ...= ...( x – x ) + H ...= ...0 + H

So if G is an abelian group with subgroup H , ...G ⁄ H ...with the operation ⊕ is called the quotient group of G by H .

Example: ...Consider the set of integers (positive, negative, and zero), Z, under addition. This is a group. Choose any integer ...n , ...then put ...H ...= ...nZ ...= ...{ n z | z ∊ Z } ...= ...{ 0, ±1n , ±2n , ⋯ } . nZ < Z , ...since n x – n y ...= ...n ( x – y ) .

The equivalence relation: ...x ∼ y ...⇔ ...( x – y ) ∊ nZ ...⇔ ...( x – y ) = n z , some z ∊ Z , ...we write ...x ≡ y ( mod n ) . ...The cosets defined by this relation are of the form ...x + nZ ...= ...{ x + n z | z ∊ Z } ...= ...{ x , x ± n , x ± 2n , ⋯ } .

Write ...Z_n ...:= ...Z ⁄ nZ ...= ...{ x + nZ } . ...Note that ...[ wn ] ...= ...[ 0 ] , ...any w ∊ Z , ...so ...Z_n ...= ...{ [ 0 ] , [ 1 ] , ⋯ , [ n – 1 ] } , ...which is written as ...{ 0 , 1 , ⋯ , n – 1 | n ≡ 0 }

Define ⊕ on Z_n by: ......x + nZ ...⊕ ...y + nZ ...= ...x + y + nZ , ...then (Z_n , ⊕ ) is isomorphic to the cyclic group of order n .

For illustrative purposes consider ...n = 3 . H ...= ...3Z ...= ...{ ⋯ –6, –3, 0, 3, 6, ⋯ }, ...x ∼ y ...⇔ ...x – y is divisible by 3, and we have

• 0 + 3Z ...= ...{ ⋯ –6, –3, 0, 3, 6, ⋯ },
• 1 + 3Z ...= ...{ ⋯ –5, –2, 1, 4, 7, ⋯ },
• 2 + 3Z ...= ...{ ⋯ –4, –1, 2, 5, 8, ⋯ },

If we just write ...0 ...for ...0 + 3Z ...etc. then we have the "addition" table:

+ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1

Addition table for (quotient) group ...Z ⁄ 3Z

 

4 Canonical Mapping

The mapping ...f : G → G ⁄ H ...given by ...f ( x ) ...= ...x + H , ...is called the canonical mapping .

f ...is a (surjective) homomorphism ...[ f ( x ) + f ( y ) ...= ...( x + H ) ⊕ ( y + H ) ...= ...( x + y ) + H ...= ...f ( x + y )] , ...so ...G ⁄ H ...is also abelian.

 

5 Kernel And Image

If G and H are abelian groups, and ...f : G → H ...is a homomorphism, then ...G ⁄ ker f ...≅ ...Im f ...(isomorphism) , ...where the isomorphism ...φ : G ⁄ ker f → Im f ...is given by ...φ ( x + ker f ) ...= ...f ( x ) .

... ———— ... G ... ———— ...   canon- ical | | ↓ ...   ............ | | ↓ ...f |Im f   ... | | ↓ ...f G ⁄ ker f ... ———→ φ Im f ... ———→ ⊆ H

φ is well-defined: ...x + ker f = x' + ker f ...⇒ ...x – x' ∊ ker f ...⇒ ...f ( x – x' ) = 0 ...⇒ ...f ( x ) = f ( x' ) ...so ...φ ( x + ker f ) = φ ( x' + ker f ) . φ is a homomorphism: ...φ ( ( x + ker f ) ⊕ ( y + ker f ) ) ...= ...φ ( ( x + y + ker f ) ) ...= ...f ( x + y ) ...= ...f ( x ) + f (y ) ...= ...φ ( x + ker f ) + φ ( y + ker f ) φ is injective: ...( x + ker f ) ≠ ( y + ker f ) ...⇒ ...x – y ∉ ker f ...⇒ ...f ( x – y ) ≠ 0 ...⇒ ...f ( x ) ≠ f ( y ) ...i.e. ...φ ( x + ker f ) ≠ φ ( y + ker f ) φ is surjective: ...y ∊ Im f ...⇒ ...y = f ( x ) = φ ( x + ker f ) , ...some x ∊ G .

6 Kernel Of Injective Homomorphisms

If G and H are any two groups, and ...f : G → H ...is a homomorphism then ......f injective ...⇔ ...ker f = { 1_G }

f injective ...⇒ ...f ( x ) ≠ f ( 1_G ) , if x ≠ 1_G .
Conversely, if ...ker f = { 1_G } , ...then ...x , y ∊ G , f ( x ) = f ( y ) ...⇔ ...f ( x y^–1 ) = f ( x ) f ( y^–1 ) = f ( x ) ( f ( y ) )^–1 = 1_H = f ( 1_G )
...⇔ ...x y^–1 = 1_G ...⇔ ...x = y .

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