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In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : where the group operation on the left hand side of the equation is that of ''G'' and on the right hand side that of ''H''. From this property, one can deduce that ''h'' maps the identity element ''eG'' of ''G'' to the identity element ''eH'' of ''H'', and it also maps inverses to inverses in the sense that : Hence one can say that ''h'' "is compatible with the group structure". Older notations for the homomorphism ''h''(''x'') may be ''x''''h'', though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that ''h''(''x'') becomes simply ''x h''. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. In areas of mathematics where one considers groups endowed with additional structure, a ''homomorphism'' sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous. == Intuition == The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function ''h'' : ''G'' → ''H'' is a group homomorphism if whenever ''a'' ∗ ''b'' = ''c'' we have ''h''(''a'') ⋅ ''h''(''b'') = ''h''(''c''). In other words, the group ''H'' in some sense has a similar algebraic structure as ''G'' and the homomorphism ''h'' preserves that. 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「group homomorphism」の詳細全文を読む スポンサード リンク
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