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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.

See also Permutation group; Transformation group.

How to Cite This Entry:
Transformation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article