Table of contents |

2 Examples 3 Properties |

Formally, *G*/*N* is the set of all left cosets of *N* in *G*, i.e.

Generally, if two subsets *S* and *T* of *G* are given, we define their product as:

(*aN*)(*bN*) = (*aN*)(*Nb*) = *a*((*NN*)*b*) = *a*(*Nb*) = *a*(*bN*) = (*ab*)*N*

-- which establishes closure. From the same calculation, it follows that *eN*=*N* is the identity element of *G*/*N* and that *a*^{-1}*N* is the inverse of *aN*.

Consider the group of integers **Z** (using addition as operation) and the subgroup 2**Z** consisting of all even integers. This is a normal subgroup because **Z** is abelian. There are only two cosets, the even and the odd numbers, and **Z** / 2**Z** is therefore isomorphic to the cyclic group with two elements.

As another abelian example, consider the group of real numbers **R** (again with addition) and the subgroup **Z** of integers.
The cosets of **Z** in **R** are of the form *a* + **Z \', with 0 ≤ a < 1 a real number. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The factor group **

If *G* is the group of invertible 3-by-3 real matrices, and *N* is the subgroup of 3-by-3 real matrices with determinant 1, then *N* is normal in *G* (since it is the kernel of the group homomorphism det), and *G*/*N* is isomorphic to the multiplicative group of non-zero real numbers.

Trivially, *G*/*G* is isomorphic to the group of order 1, and *G*/{e} is isomorphic to *G*.

There is a "natural" surjective group homomorphism π : *G* → *G*/*N*, sending each element *g* of *G* to the coset of *N* to which *g* belongs, that is: π(*g*) = *gN*. The application π is sometimes called *canonical projection*. Its kernel is *N*.

There is a bijective correspondence between the subgroups of *G* that contain *N* and the subgroups of *G*/*N*; if *H* is a subgroup of *G* containing *N*, then the corresponding subgroup of *G*/*N* is π(*H*). This correspondence holds for normal subgroups of *G* and *G*/*N* as well, and is formalized in the lattice theorem.

Several important properties of factor groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.

If *G* is abelian, nilpotent or solvable, then so is *G*/*N*.

If *G* is cyclic or finitely generated, then so is *G*/*N*.

Every group is isomorphic to a group of the form *F*/*N*, where *F* is a free group and *N* is a normal subgroup of *F*.