Quotient Groups

 
 

Cosets

Let ~G be a group with subgroup ~H. _ Consider the relation _ ~x ~ ~y _ <=> _ ~y = ~h ~x , some ~h &in. ~H. _ [ equivalently _ ~y ~x^{-1} &in. ~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 &in. ~G , the ~#{right coset} of ~x in ~H , _ _ ~H ~x _ #:= _ [ ~x ] _ = _ \{ ~y &in. ~G | ~y = ~h ~x , some ~h &in. ~H \} , _ i.e. the equivalence class of ~x in the above relation.
Analagously, the ~#{left coset} of ~x in ~H , _ _ ~x ~H _ #:= _ \{ ~z &in. ~G | ~z = ~x ~h , some ~h &in. ~H \} _ ( using the relation _ ~x ~ ~z _ <=> _ ~z = ~x ~h , some ~h &in. ~H. )

In general _ ~x ~H _ != _ ~H ~x .

The set of right cosets in ~H is denoted _ ~H ~G _ #:= _ \{ ~H ~x | ~x &in. ~G \} , _ and the set of left cosets in ~H is denoted _ ~G ./ ~H _ #:= _ \{ ~x ~H | ~x &in. ~G \}

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 = \{ ~h_1 , ... , ~h_~n \} , _ ~n distinct ~h_~i, _ then _ ~H ~x = \{ ~h_1 ~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 | .

 

Quotient Group

If ~G is an abelian group ( with group operation '+' ) then _ ~x ~ ~y _ if _ ~x - ~y &in. ~H , _ and the left and right cosets coincide, _
i.e. _ ~x + ~H = ~H + ~x , _ &forall. ~x . _ ~H ~G _ = _ ~G ./ ~H _ = _ \{ ~x + ~H | ~x &in. ~G\}

Define the operation _ &oplus. #: ( ~G ./ ~H ) # ( ~G ./ ~H ) -> ( ~G ./ ~H ) _ by _ ( ~x + ~H ) &oplus. ( ~y + ~H ) _ = _ ( ~x + ~y ) + ~H .

With this operation _ ~G ./ ~H _ is a group:

  • associative: _ ( ( ~x + ~H ) &oplus. ( ~y + ~H ) ) &oplus. ( ~z + ~H ) _ = _ ( ( ~x + ~y ) + ~H ) &oplus. ( ~z + ~H )
    _ _ _ = _ ( ( ~x + ~y + ~z) + ~H ) _ = _ ( ~x + ~H ) &oplus. ( ( ~y + ~z ) + ~H ) _ = _ ( ~x + ~H ) &oplus. ( ( ~y + ~H ) &oplus. ( ~z + ~H ) ) .
  • identity: _ ( 0 + ~H ) &oplus. ( ~x + ~H ) _ = _ ( 0 + ~x ) + ~H _ = _ ~x + ~H
  • inverse: _ ( ~x + ~H ) &oplus. ( -~x + ~H ) _ = _ ( ~x - ~x^ ) + ~H _ = _ 0 + ~H

So if ~G is an abelian group with subgroup ~H , _ ~G ./ ~H _ with the operation &oplus. 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 _ = _ ~n#Z _ = _ \{ ~n ~z | ~z &in. #Z \} _ = _ \{ 0, +-1~n , +-2~n , ... \} . ~n#Z < #Z , _ since ~n ~x - ~n ~y _ = _ ~n ( ~x - ~y ) .

The equivalence relation: _ ~x ~ ~y _ <=> _ ( ~x - ~y ) &in. ~n#Z _ <=> _ ( ~x - ~y ) = ~n ~z , some ~z &in. #Z , _ we write _ ~x == ~y ( mod ~n ) . _ The cosets defined by this relation are of the form _ ~x + ~n#Z _ = _ \{ ~x + ~n ~z | ~z &in. #Z \} _ = _ \{ ~x , ~x +- ~n , ~x +- 2~n , ... \} .

Write _ #Z_~n _ #:= _ #Z ./ ~n#Z _ = _ \{ ~x + ~n#Z \} . _ Note that _ [ ~w~n ] _ = _ [ 0 ] , _ any ~w &in. #Z , _ so _ #Z_~n _ = _ \{ [ 0 ] , [ 1 ] , ... , [ ~n - 1 ] \} , _ which is written as _ \{ 0 , 1 , ... , ~n - 1 | ~n == 0 \}

Define &oplus. on #Z_~n by: _ _ ~x + ~n#Z _ &oplus. _ ~y + ~n#Z _ = _ ~x + ~y + ~n#Z , _ then (#Z_~n , &oplus. ) is isomorphic to the cyclic group of order ~n .

For illustrative purposes consider _ ~n = 3 . ~H _ = _ 3#Z _ = _ \{ ... -6, -3, 0, 3, 6, ... \}, _ ~x ~ ~y _ <=> _ ~x - ~y is divisible by 3, and we have

  • 0 + 3#Z _ = _ \{ ... -6, -3, 0, 3, 6, ... \},
  • 1 + 3#Z _ = _ \{ ... -5, -2, 1, 4, 7, ... \},
  • 2 + 3#Z _ = _ \{ ... -4, -1, 2, 5, 8, ... \},

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

array{ + ,#0,#1,#2,/#0,0,1,2/#1,1,2,0/#2,2,0,1}

   Addition table for (quotient) group _ #Z ./ 3#Z

 

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 ) &oplus. ( ~y + ~H ) _ = _ ( ~x + ~y ) + ~H _ = _ ~f ( ~x + ~y )] , _ so _ ~G ./ ~H _ is also abelian.

 

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 _ &phi. #: ~G ./ ker ~f -> Im ~f _ is given by _ &phi. ( ~x + ker ~f ) _ = _ ~f ( ~x ) .

  zDgrmHzLine{ _ , _ } ~G zDgrmHzLine{ _ , _ }  
zDgrmDown{canon-&br.ical, _ }   zDgrmDown{ _ _ _ _ , _ ~f |_{Im ~f}}   zDgrmDown{ _ , _ ~f}
~G ./ ker ~f zDgrmRight{ _ ,&phi.} Im ~f zDgrmRight{ _ ,&subseteq.} ~H
 

&phi. is well-defined: _ ~x + ker ~f = ~x' + ker ~f _ => _ ~x - ~x' &in. ker ~f _ => _ ~f ( ~x - ~x' ) = 0 _ => _ ~f ( ~x ) = ~f ( ~x' ) _ so _ &phi. ( ~x + ker ~f ) = &phi. ( ~x' + ker ~f ) .

&phi. is a homomorphism: _ &phi. ( ( ~x + ker ~f ) &oplus. ( ~y + ker ~f ) ) _ = _ &phi. ( ( ~x + ~y + ker ~f ) ) _ = _ ~f ( ~x + ~y ) _ = _ ~f ( ~x ) + ~f (~y ) _ = _ &phi. ( ~x + ker ~f ) + &phi. ( ~y + ker ~f )

&phi. is injective: _ ( ~x + ker ~f ) != ( ~y + ker ~f ) _ => _ ~x - ~y &nin. ker ~f _ => _ ~f ( ~x - ~y ) != 0 _ => _ ~f ( ~x ) != ~f ( ~y ) _ i.e. _ &phi. ( ~x + ker ~f ) != &phi. ( ~y + ker ~f )

&phi. is surjective: _ ~y &in. Im ~f _ => _ ~y = ~f ( ~x ) = &phi. ( ~x + ker ~f ) , _ some ~x &in. ~G .

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 &in. ~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 .