Quotient Groups

Definition (Kernel) If is a homomorphism then the kernel of is defined by

Example (Kernel) (i) For any positive integer define by for each Then for all so that is a homomorphism and
    (ii) Define by for The for all Thus is a homormorphism and
    (iii)Let and be groups and be their direct product. Then defined by is a homomorphism from onto because
    


for all Notice that This homormprohism is called the natural projection of onto

Definition (Normal Subgroup) A subgroup of is called a normal subgroup of when for all and all and is denoted by

Example (Normal Subgroup) (i) Every subgroup of an Abelian group is a normal subgroup.
    (ii) The center of a group is always normal.
    (iii) The alternating group is even permutation is a normal subgroup of For example is a normal subgroup of Notice that is not a normal subgroup of because
    (iv) The subgroup of rotations in is normal in
    (v) The group of matrices with determinant 1 is a normal subgroup of the group of matrices with nonzero determinant.

Proposition (Kernel) If is a homomorphism then,

    (i)   is a subgroup of
    
    (ii) is one-to-one if and only if and
    
    (iii)
    
    Proof. (i) The kernel of is not empty because If then and so Therefore, and so is a subgroup of
    (ii) Suppose is one-to-one. Clearly, If then and since it follows that Conversely, suppose and Then   and so Thus and therefore,
    (iii) By (i) is a subgroup of . Let and Then and so Thus, and so as desired.

Proposition (Quotient Subgroup) Let be a normal subgroup of Then is a group where denotes the set of all right cosets of in and the group operation is defined by The group is called the quotient group of by

    Proof. To show that this operation is well-defined on let and We must show From we have for some Similarly,   for some Then We would like to switch the and in the last product and we can (almost) since which means that and so Thus,   becomes Therefore, The identity is and every element of has an inverse because is the inverse for Finally the operation is associative because

Proposition (Normal Subgroups are Kernels of Homomorphisms) If is a normal subgroup of  a group then defined by for each is a homomorphism of onto and

    Proof. Because is a partition of is a mapping that is onto. Since is a homomorphism. The last statement follows from, if then   if and only if  

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