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In group theory, a word is any written product of group elements and their inverses. For example, if ''x'', ''y'' and ''z'' are elements of a group ''G'', then ''xy'', ''z''−1''xzz'' and ''y''−1''zxx''−1''yz''−1 are words in the set . Two different words may evaluate to the same value in ''G'',〔for example, fdr1 and r1fc in the group of square symmetries〕 or even in every group.〔for example, ''xy'' and ''xzz''−1''y''〕 Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory. ==Definition== Let ''G'' be a group, and let ''S'' be a subset of ''G''. A word in ''S'' is any expression of the form : where ''s''1,...,''sn'' are elements of ''S'' and each ''εi'' is ±1. The number ''n'' is known as the length of the word. Each word in ''S'' represents an element of ''G'', namely the product of the expression. By convention, the identity (unique)〔Uniqueness of identity element and inverses〕 element can be represented by the empty word, which is the unique word of length zero. 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「Word (group theory)」の詳細全文を読む スポンサード リンク
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