if G is a group with an abelian quotient group G/N,then N is abelian is it true
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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using anequivalence relation that preserves the group structure. For example, the cyclic group of addition modulo n can be obtained from theintegers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as acongruence class) as a single entity. It is part of the mathematical field known as group theory.
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. 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 theimage of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism φ: G → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller