Definition:Quotient Group

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Definition

Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.


Then the left coset space $G / N$ is a group, where the group operation is defined as:

$\paren {a N} \paren {b N} = \paren {a b} N$


$G / N$ is called the quotient group of $G$ by $N$.


It is proven to be a group in Quotient Group is Group.


Also known as

A quotient group is also known as a factor group.


Motivation

In Kernel is Normal Subgroup of Domain it was shown that the kernel of a group homomorphism is a normal subgroup of its domain.

In this result it has been shown that every normal subgroup is a kernel of at least one group homomorphism of the group of which it is the subgroup.

We see that when a subgroup is normal, its cosets make a group using the group operation defined as in this result.

However, it is not possible to make the left or right cosets of a non-normal subgroup into a group using the same sort of group operation.

Otherwise there would be a group homomorphism with that subgroup as the kernel, and we have seen that this can not be done unless the subgroup is normal.


Also see

It follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.

Thus we do not need to distinguish between the left quotient group and the right quotient group - the two are one and the same.

  • Results about quotient groups can be found here.


Historical Note

The idea of a quotient group appeared in the work of Marie Ennemond Camille Jordan in the $1860$s.

However, the modern formulation using cosets did not appear till the work of Otto Ludwig Hölder in $1889$, fairly late on in the history of group theory.


Sources