QUOTIENT GROUP

 A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of  and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory.

For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written ${\displaystyle �/�}$, where ${\displaystyle �}$ is the original group and ${\displaystyle �}$ is the normal subgroup. (This is pronounced ${\displaystyle �mod�}$, where ${\displaystyle \text{mod}}$ is short for modulo.)

Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of ${\displaystyle �}$. Specifically, the image of ${\displaystyle �}$ under a homomorphism ${\displaystyle �:�\to �}$ is isomorphic to ${\displaystyle �/\ker (�)}$ where ${\displaystyle \ker (�)}$ denotes the kernel of ${\displaystyle �}$.

The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.

For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.

Definition and illustration[edit]

Given a group ${\displaystyle �}$ and a subgroup ${\displaystyle �}$, and a fixed element ${\displaystyle �\in �}$, one can consider the corresponding left coset: ${\displaystyle ��:=\left\{�ℎ:ℎ\in �\right\}}$. Cosets are a natural class of subsets of a group; for example consider the abelian group G of integers, with operation defined by the usual addition, and the subgroup ${\displaystyle �}$ of even integers. Then there are exactly two cosets: ${\displaystyle 0+�}$, which are the even integers, and ${\displaystyle 1+�}$, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).

For a general subgroup ${\displaystyle �}$, it is desirable to define a compatible group operation on the set of all possible cosets, ${\displaystyle \left\{��:�\in �\right\}}$. This is possible exactly when ${\displaystyle �}$ is a normal subgroup, see below. A subgroup ${\displaystyle �}$ of a group ${\displaystyle �}$ is normal if and only if the coset equality ${\displaystyle ��=��}$ holds for all ${\displaystyle �\in �}$. A normal subgroup of ${\displaystyle �}$ is denoted ${\displaystyle �}$.

Definition[edit]

Let ${\displaystyle �}$ be a normal subgroup of a group ${\displaystyle �}$ . Define the set ${\displaystyle �/�}$ to be the set of all left cosets of ${\displaystyle �}$ in ${\displaystyle �}$ . That is, ${\displaystyle �/�=\left\{��:�\in �\right\}}$. Since the identity element ${\displaystyle �\in �}$, ${\displaystyle �\in ��}$. Define a binary operation on the set of cosets, ${\displaystyle �/�}$, as follows. For each ${\displaystyle ��}$ and ${\displaystyle ��}$ in ${\displaystyle �/�}$, the product of ${\displaystyle ��}$ and ${\displaystyle ��}$, ${\displaystyle (��)(��)}$, is ${\displaystyle (��)�}$. This works only because ${\displaystyle (��)�}$ does not depend on the choice of the representatives, ${\displaystyle �}$ and ${\displaystyle �}$, of each left coset, ${\displaystyle ��}$ and ${\displaystyle ��}$. To prove this, suppose ${\displaystyle ��=��}$ and ${\displaystyle ��=��}$ for some ${\displaystyle �,�,�,�\in �}$. Then

$(��)�=�(��)=�(��)=�(��)=(��)�=(��)�=�(��)=�(��)=(��)�$.

This depends on the fact that N is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on G/N.

To show that it is necessary, consider that for a subgroup ${\displaystyle �}$ of ${\displaystyle �}$, we have been given that the operation is well defined. That is, for all ${\displaystyle ��=��}$ and ${\displaystyle ��=��}$, for ${\displaystyle �,�,�,�\in �,(��)�=(��)�}$.

Let ${\displaystyle �\in �}$ and ${\displaystyle �\in �}$. Since ${\displaystyle ��=��}$, we have ${\displaystyle ��=(��)�=(��)(��)=(��)(��)=(��)�}$.

Now, ${\displaystyle ��=(��)�\iff �=({�}^{-1}��)�\iff {�}^{-1}��\in �,\mathrm{\forall }�\in �}$ and ${\displaystyle �\in �}$.

Hence ${\displaystyle �}$ is a normal subgroup of ${\displaystyle �}$ .

It can also be checked that this operation on ${\displaystyle �/�}$ is always associative, ${\displaystyle �/�}$ has identity element ${\displaystyle �}$, and the inverse of element ${\displaystyle ��}$ can always be represented by ${\displaystyle {�}^{-1}�}$. Therefore, the set ${\displaystyle �/�}$ together with the operation defined by ${\displaystyle (��)(��)=(��)�}$ forms a group, the quotient group of ${\displaystyle �}$ by ${\displaystyle �}$.

Due to the normality of ${\displaystyle �}$, the left cosets and right cosets of ${\displaystyle �}$ in ${\displaystyle �}$ are the same, and so, ${\displaystyle �/�}$ could have been defined to be the set of right cosets of ${\displaystyle �}$ in ${\displaystyle �}$ .

Example: Addition modulo 6[edit]

For example, consider the group with addition modulo 6: ${\displaystyle �=\left\{0,1,2,3,4,5\right\}}$. Consider the subgroup ${\displaystyle �=\left\{0,3\right\}}$, which is normal because ${\displaystyle �}$ is abelian. Then the set of (left) cosets is of size three:

${\displaystyle �/�=\left\{�+�:�\in �\right\}=\left\{\left\{0,3\right\},\left\{1,4\right\},\left\{2,5\right\}\right\}=\left\{0+�,1+�,2+�\right\}}$.

The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.