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In group theory, a branch of mathematics, the term ''order'' is used in two unrelated senses: * The order of a group is its cardinality, i.e., the number of elements in its set. Also, the order, sometimes period, of an element ''a'' of a group is the smallest positive integer ''m'' such that (where ''e'' denotes the identity element of the group, and ''a''''m'' denotes the product of ''m'' copies of ''a''). If no such ''m'' exists, ''a'' is said to have infinite order. * The ordering relation of a partially or totally ordered group. This article is about the first notions. The order of a group ''G'' is denoted by ord(''G'') or and the order of an element ''a'' is denoted by ord(''a'') or . ==Example== Example. The symmetric group S3 has the following multiplication table. : This group has six elements, so ord(S3) = 6. By definition, the order of the identity, ''e'', is 1. Each of ''s'', ''t'', and ''w'' squares to ''e'', so these group elements have order 2. Completing the enumeration, both ''u'' and ''v'' have order 3, for ''u''2 = ''v'' and ''u''3 = ''vu'' = ''e'', and ''v''2 = ''u'' and ''v''3 = ''uv'' = ''e''. 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「Order (group theory)」の詳細全文を読む スポンサード リンク
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