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Concept# Quotient group

Summary

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).

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Normal subgroup

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group.

Equivalence class

In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set and an equivalence relation on the of an element in denoted by is the set of elements which are equivalent to It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of by and is denoted by .

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