Definition:Quotient Group

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 product is defined as:

$\left({a N}\right) \left({b N}\right) = \left({a b}\right) N$

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

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

Also known as

A quotient group is also known as a factor group.

Also see

• Normal Subgroup Equivalent Definitions: $(2)$: the set of left cosets is equal to the set of right cosets.

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.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.7$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 11$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.10$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Theorem $3$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 47$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 50$
• John F. Humphreys: A Course in Group Theory (1996): $\S 7$: Proposition $7.11$