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 ${\displaystyle n}$ 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 ⁠${\displaystyle G\,/\,N}$⁠, where ${\displaystyle G}$ is the original group and ${\displaystyle N}$ is the normal subgroup. This is read as '⁠${\displaystyle G{\bmod {N}}}$⁠', where ${\displaystyle {\text{mod}}}$ is short for modulo. (The notation ⁠${\displaystyle G\,/\,H}$⁠ should be interpreted with caution, as some authors (e.g., Vinberg) use it to represent the left cosets of ${\displaystyle H}$ in ${\displaystyle G}$ for any subgroup ${\displaystyle H}$, even though these cosets do not form a group if ${\displaystyle H}$ is not normal in ⁠${\displaystyle G}$⁠. Others (e.g., Dummit and Foote) use this notation to refer only to the quotient group, with the appearance of this notation implying that ${\displaystyle H}$ is normal in ⁠${\displaystyle G}$⁠.)

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 ${\displaystyle G}$ under a homomorphism is always isomorphic to a quotient of ⁠${\displaystyle G}$⁠. Specifically, the image of ${\displaystyle G}$ under a homomorphism ${\displaystyle \varphi :G\rightarrow H}$ is isomorphic to ${\displaystyle G\,/\,\ker(\varphi )}$ where ${\displaystyle \ker(\varphi )}$ denotes the kernel of ⁠${\displaystyle \varphi }$⁠.

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.

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

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

Let ${\displaystyle N}$ be a normal subgroup of a group ⁠${\displaystyle G}$⁠. Define the set ${\displaystyle G\,/\,N}$ to be the set of all left cosets of ${\displaystyle N}$ in ⁠${\displaystyle G}$⁠. That is, ⁠${\displaystyle G\,/\,N=\left\{aN:a\in G\right\}}$⁠.

Since the identity element ⁠${\displaystyle e\in N}$⁠, ⁠${\displaystyle a\in aN}$⁠. Define a binary operation on the set of cosets, ⁠${\displaystyle G\,/\,N}$⁠, as follows. For each ${\displaystyle aN}$ and ${\displaystyle bN}$ in ⁠${\displaystyle G\,/\,N}$⁠, the product of ${\displaystyle aN}$ and ⁠${\displaystyle bN}$⁠, ⁠${\displaystyle (aN)(bN)}$⁠, is ⁠${\displaystyle (ab)N}$⁠. This works only because ${\displaystyle (ab)N}$ does not depend on the choice of the representatives, ${\displaystyle a}$ and ⁠${\displaystyle b}$⁠, of each left coset, ${\displaystyle aN}$ and ⁠${\displaystyle bN}$⁠. To prove this, suppose ${\displaystyle xN=aN}$ and ${\displaystyle yN=bN}$ for some ⁠${\displaystyle x,y,a,b\in G}$⁠. Then