In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical. A common way of specifying non-canonical actions is to describe a homomorphism on a point is assumed to be identical to the action of its image

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dbo:abstract
• In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. Some groups can be interpreted as acting on spaces in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another element of the set. More generally, symmetry groups such as the homeomorphism group of a topological space or the general linear group of a vector space, as well as their subgroups, also admit canonical actions. For other groups, an interpretation of the group in terms of an action may have to be specified, either because the group does not act canonically on any space or because the canonical action is not the action of interest. For example, we can specify an action of the two-element cyclic group on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical. A common way of specifying non-canonical actions is to describe a homomorphism from a group G to the group of symmetries of a set X. The action of an element on a point is assumed to be identical to the action of its image on the point . The homomorphism is also frequently called the "action" of G, since specifying is equivalent to specifying an action. Thus, if G is a group and X is a set, then an action of G on X may be formally defined as a group homomorphism from G to the symmetric group of X. The action assigns a permutation of X to each element of the group in such a way that: * the identity element of G is assigned the identity transformation of X; * any product gk of two elements of G is assigned the composition of the permutations assigned to g and k. If X has additional structure, then is only called an action if for each , the permutation preserves the structure of X. The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Because of this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields. (en)
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• 12781 (xsd:integer)
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• 745023622 (xsd:integer)
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• March 2015
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• p/a010550
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• The isometries of a space are a subgroup of the affine group of that space, but not an affine group in themselves
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• Action of a group on a manifold
• Group Action
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• GroupAction
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http://purl.org/linguistics/gold/hypernym
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• In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. on the finite set by specifying that 0 (the identity element) sends , and that 1 sends . This action is not canonical. A common way of specifying non-canonical actions is to describe a homomorphism on a point is assumed to be identical to the action of its image (en)
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• Group action (en)
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