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 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 , where is the original group and is the normal subgroup. (This is pronounced , where 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 of any group G under a homomorphism is always isomorphic to a quotient of . Specifically, the image of under a homomorphism is isomorphic to where denotes the kernel of . 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 , quotient groups are examples of quotient objects, which are to subobjects. Given a group and a subgroup , and a fixed element , one can consider the corresponding left coset: . 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 of even integers. Then there are exactly two cosets: , which are the even integers, and , which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).