A mapping of a set (in general endowed with some structure) into itself. The image of an element under the transformation is denoted by , , , or . The set of all transformations of a set into itself forms a transformation semi-group with respect to multiplication (composition), which is called the symmetric transformation semi-group on . The invertible elements of this semi-group are called permutations (cf. Permutation of a set). All permutations on a set form a subgroup of the symmetric semi-group — the symmetric group.
Transformation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Transformation&oldid=12741