Group action

This article is about the mathematical concept. For the sociology term, see group action (sociology).

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In mathematics, a symmetry group describes all symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case—it is the same as a group action of G on an ordered basis of a vector space.

Related Topics:
Mathematics - Symmetry group - Symmetries - Group - Bijective - Permutation group - Vector space - Linear transformation - Permutation - Group representation - Permutation matrices - Basis of a vector space

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