Quotient group

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

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.

⁠ should be interpreted with caution, as some authors (e.g., Vinberg[1]) use it to represent the left cosets of

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

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.

of integers, with operation defined by the usual addition, and the subgroup

⁠, it is desirable to define a compatible group operation on the set of all possible cosets, ⁠

Define a binary operation on the set of cosets, ⁠

It still remains to be shown that this condition is not only sufficient but necessary to define the operation on ⁠

Then the set of (left) cosets is of size three: 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.

⁠, the group structure is used to form a natural "regrouping".

Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.

The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group ⁠

⁠, shown on the picture on the right as colored balls with the number at each point giving its complex argument.

made of the fourth roots of unity, shown as red balls.

This normal subgroup splits the group into three cosets, shown in red, green and blue.

Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1.

is isomorphic to the multiplicative group of non-zero real numbers.

The Paillier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of

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

is abelian, nilpotent, solvable, cyclic or finitely generated, then so is ⁠

This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups.

is the smallest prime number dividing the order of a finite group, ⁠

is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup of ⁠

has the structure of a fiber bundle (specifically, a principal ⁠

is not closed then the quotient space is not a T1-space (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a Hausdorff space.

of left cosets is not a group, but simply a differentiable manifold on which