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...
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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...
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