In this article, we will learn about the definition of quotient groups along with their examples, properties, and a few solved problems.

Table of Contents

## Definition of Quotient Group

Let H be a normal subgroup of a group G. Consider the set of all left cosets of H in G, that is, {aH: a∈G}. This set forms a group under the composition aH*bH = abH for all a,b ∈ G, denoted by G/H. This group (G/H, *) is called the quotient group of G by H.

If H is a normal subgroup of G, then the set G/H containing of all left cosets of H in G forms a group, called the quotient group of G by H. The quotient groups are also known as the factor groups. |

## Examples of Quotient Groups

Below are a few examples of quotient groups (or factor groups).

1. Let G=(Z_{6}, +) be the additive group and H={[0], [3]}. Then H is a normal subgroup of G and the quotient group G/H contains three elements H, [0]+H, [1]+H.

2. Let G=(Z, +) be the additive group of integers, and H=(2Z, +) is a normal subgroup of G. Then the set G/H consisting of left cosets of H in G forms the quotient group having two elements H and 1+H.

## Properties of Quotient Groups

A quotient group satisfies the following properties.

- The trivial group {e
_{G}} is always a normal subgroup of G and the corresponding quotient group G/{e_{G}} is the group G itself. - The quotient group of G by G, that is, G/G is the trivial group.
- The identity element in a quotient group G/H is the normal subgroup H itself.
- The order of a quotient group G/H is equal to the order of G divided by the order of H. In other words, |G/H| = |G|/|H|.
- If H is a subgroup of an abelian group G, then the quotient group G/H must be also abelian. But, not that the converse is not always true. For example, let G=S
_{3}be the permutation group on three objects and H=A_{3}be its alternating group. As G/H =2, the quotient group G/H is abelian, but we know that G is not abelian. - If H is a subgroup of a cyclic group G, then the quotient group G/H is cyclic. But, not that the converse is not always true. In the above example, G/H=S
_{3}/A_{3}is cyclic but G=S_{3}is not cyclic. - If G is a non-abelian group with center Z(G), then the factor group G/Z(G) is non-cyclic.
- If G is a solvable group, then every quotient group G/H is also solvable.
- The quotient group G/H is nilpotent if G is so.

## Solved Problems on Quotient Group

**Question 1:** Prove that the symmetric group S_{3} has a trivial center.

*Answer:*

Let Z denote the center of S_{3}. As S_{3} is non-abelian, we conclude that Z≠S_{3}.

We know that Z is a normal subgroup of S_{3}. Thus, by Lagrange’s theorem, the order |Z| must be a divisor of |S_{3}|=6. As |Z|≠6, we deduce that

|Z| = 1, 2, or 3.

If |Z|=3, then |S_{3}/Z|=2. Hence, S_{3}/Z is cyclic but this is an impossibility by the above property 7. Therefore, |Z| cannot be equal to 3.

If |Z|=2, then |S_{3}/Z|=3. Hence, S_{3}/Z is cyclic. But, again this is an impossibility by the above property 7. Therefore, |Z| cannot be equal to 2.

Therefore, we obtain the conclusion that |Z|=1. Hence, S_{3} has the trivial center.

**Related Topics:**

**Group Theory: Definition, Examples, Orders, Types, Properties, Applications**

**Abelian Group: Definition, Properties, Examples**

**Cyclic Group: Definition, Orders, Properties, Examples**

## FAQs on Quotient Group

**Q1: What is quotient group with example?**

Answer: The set of all left cosets of a normal subgroup H in a group G forms a group and this group, denoted by G/H, is called the quotient group of G by H. For example, Z/2Z is a quotient group with two elements 2Z, 2Z+1.

**Q2: What is the order of a quotient group?**

Answer: The order of a quotient group G/H is given by the formula |G/H| = |G|/|H|, that is, the order of a quotient group is equal to the order of the group divided by the order of the corresponding normal subgroup.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.