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What (who) is properly discontinuous - definition
![In the [[compound of five tetrahedra]], the symmetry group is the (rotational) icosahedral group ''I'' of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) [[tetrahedral group]] ''T'' of order 12, and the orbit space ''I''/''T'' (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset ''gT'' corresponds to the tetrahedron to which ''g'' sends the chosen tetrahedron.](https://commons.wikimedia.org/wiki/Special:FilePath/Compound of five tetrahedra.png?width=200 "In the [[compound of five tetrahedra]], the symmetry group is the (rotational) icosahedral group ''I'' of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) [[tetrahedral group]] ''T'' of order 12, and the orbit space ''I''/''T'' (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset ''gT'' corresponds to the tetrahedron to which ''g'' sends the chosen tetrahedron.")
![The [[cyclic group]] ''C''<sub>3</sub> consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.](https://commons.wikimedia.org/wiki/Special:FilePath/Group action on equilateral triangle.svg?width=200 "The [[cyclic group]] ''C''<sub>3</sub> consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.")
Wikipedia
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K.
The symmetric group Sn acts on any set with n elements by permuting the elements of the set. Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality.
