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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937201.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937202.png" /> (in general endowed with some structure) into itself. The image of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937203.png" /> under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937204.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937207.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937208.png" />. The set of all transformations of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t0937209.png" /> into itself forms a transformation semi-group with respect to multiplication (composition), which is called the symmetric [[Transformation semi-group|transformation semi-group]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t09372010.png" />. The invertible elements of this semi-group are called permutations (cf. [[Permutation of a set|Permutation of a set]]). All permutations on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093720/t09372011.png" /> form a subgroup of the symmetric semi-group — the [[Symmetric group|symmetric group]].
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A mapping $u$ of a set $X$ (in general endowed with some structure) into itself. The image of an element $x \in X$ under the transformation $u$ is denoted by $u(x)$, $ux$, $x u$ or $x^u$. The set of all transformations of a set $X$ into itself forms a [[monoid]] with respect to multiplication (composition), with the [[identity map]] as identity element, which is called the symmetric [[transformation semi-group]] on $X$. The invertible elements of this semi-group are called permutations (cf. [[Permutation of a set]]). All permutations on a set $X$ form a subgroup of the symmetric semi-group — the [[symmetric group]] on $X$.
  
See also [[Permutation group|Permutation group]]; [[Transformation group|Transformation group]].
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See also [[Permutation group]]; [[Transformation group]].
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Latest revision as of 22:15, 24 September 2016

A mapping $u$ of a set $X$ (in general endowed with some structure) into itself. The image of an element $x \in X$ under the transformation $u$ is denoted by $u(x)$, $ux$, $x u$ or $x^u$. The set of all transformations of a set $X$ into itself forms a monoid with respect to multiplication (composition), with the identity map as identity element, which is called the symmetric transformation semi-group on $X$. The invertible elements of this semi-group are called permutations (cf. Permutation of a set). All permutations on a set $X$ form a subgroup of the symmetric semi-group — the symmetric group on $X$.

See also Permutation group; Transformation group.

How to Cite This Entry:
Transformation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Transformation&oldid=39298
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article