Quotient group

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In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by identifying together elements of a larger group using an equivalence relation. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity.

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 the cosets of this normal subgroup. The resulting quotient is written G / N, where G is the original group and N is the 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. Specifically, the image of G under a homomorphism φG → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.

Theoretically, the notion of a quotient group is dual to the notion of a subgroup, these being the two primary ways of forming a smaller group from a larger one. 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.

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Product of subsets of a group

In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as ST = {st : s in S and t in T}. This operation is associative and has as identity element the singleton {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation.