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In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure. 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 precisely the cosets of that normal subgroup. The resulting quotient is written , 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 is isomorphic to 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 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 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. ==Definition and illustration== Given a group ''G'' and a subgroup ''H'', and an element ''a'' in ''G'', then one can consider the corresponding left coset : ''aH'':=. Cosets are a natural class of subsets of a group; for example consider the abelian group ''G'' of integers, and the subgroup ''H'' of even integers. Then there are exactly two cosets: ''0 + H'', which are the even integers, and ''1 + H'', which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation). For a general subgroup ''H'', it is desirable to define a compatible group operation on the set of all possible cosets, . This is possible exactly when ''H'' is a normal subgroup, as we will see below. A subgroup ''N'' of a group ''G'' is normal if and only if the coset equality ''aN'' = ''Na'' holds for all ''a'' in ''G''. A normal subgroup of ''G'' is denoted . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quotient group」の詳細全文を読む スポンサード リンク
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