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Tietze transformations
In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group. These transformations are named after Heinrich Franz Friedrich Tietze who introduced them in a paper in 1908.
A presentation is in terms of ''generators'' and ''relations''; formally speaking the presentation is a pair of a set of named generators, and a set of words in the free group on the generators that are taken to be the relations. Tietze transformations are built up of elementary steps, each of which individually rather evidently takes the presentation to a presentation of an isomorphic group. These elementary steps may operate on generators or relations, and are of four kinds.
==Adding a relation==
If a relation can be derived from the existing relations then it may be added to the presentation without changing the group. Let G=〈 x | x3=1 〉 be a finite presentation for the cyclic group of order 3. Multiplying x3=1 on both sides by x3 we get x6 = x3 = 1 so x6 = 1 is derivable from x3=1. Hence G=〈 x | x3=1, x6=1 〉 is another presentation for the same group.
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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