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Examples of groups
Some elementary examples of groups in mathematics are given on Group (mathematics).
Further examples are listed here.
==Permutations of a set of three elements==
(詳細はWe can write ''xy'' for the operation "first do ''y'', then do ''x''"; so that ''ab'' is the operation RGB → RBG → BRG, which could be described as "move the first two blocks one position to the right and put the third block into the first position". If we write ''e'' for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows:
* ''e'' : RGB → RGB
* ''a'' : RGB → GRB
* ''b'' : RGB → RBG
* ''ab'' : RGB → BRG
* ''ba'' : RGB → GBR
* ''aba'' : RGB → BGR
Note that ''aa'' has the effect RGB → GRB → RGB; so we can write ''aa'' = ''e''. Similarly, ''bb'' = (''aba'')(''aba'') = ''e''; (''ab'')(''ba'') = (''ba'')(''ab'') = ''e''; so every element has an inverse.
By inspection, we can determine associativity and closure; note in particular that (''ba'')''b'' = ''aba'' = ''b''(''ab'').
Since it is built up from the basic operations ''a'' and ''b'', we say that the set ''generates'' this group. The group, called the ''symmetric group'' ''S''3, has order 6, and is non-abelian (since, for example, ''ab'' ≠ ''ba'').
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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