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In a quotient of 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.
This is pronounced " G mod N ", where "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 G.
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.
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 H of even integers.
This is possible exactly when H is a normal subgroup , see below. Let N be a normal subgroup of a group G.
This works only because ab N does not depend on the choice of the representatives, a and b , of each left coset, aN and bN. This depends on the fact that N is a normal subgroup.
To show that it is necessary, consider that for a subgroup N of G , we have been given that the operation is well defined. 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.
When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.
These are the cosets of N in G. 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.
Consider the group of integers Z under addition and the subgroup 2 Z consisting of all even integers. This is a normal subgroup, because Z is abelian.
A slight generalization of the last example. Once again consider the group of integers Z under addition. Let n be any positive integer. We will consider the subgroup n Z of Z consisting of all multiples of n.
Once again n Z is normal in Z because Z is abelian. This is a cyclic group of order n. The twelfth roots of unity , which are points on the complex unit circle , form a multiplicative abelian group G , shown on the picture on the right as colored balls with the number at each point giving its complex argument.
Consider its subgroup N 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.
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