What does group action mean?

Definitions for group action
group action

This dictionary definitions page includes all the possible meanings, example usage and translations of the word group action.

Princeton's WordNet

  1. group actionnoun

    action taken by a group of people

Wiktionary

  1. group actionnoun

    a way of describing symmetries of objects using groups

  2. group actionnoun

    A situation in which a large number of agents take action simultaneously in order to achieve a common goal; their actions are usually coordinated.

Wikipedia

  1. Group action

    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.

ChatGPT

  1. group action

    A group action is a formal way of describing the way that elements of a group (an algebraic structure) interact with another set. More specifically, it is a function or operation which combines an element from a group and an element from this set, following two key axioms. These axioms are: 1. For any element "x" in the set, the identity element of the group, commonly represented as "e", when combined with "x", leaves "x" unchanged. 2. For any two elements "g" and "h" of the group and any element "x" of the set, first combining "g" with "h" and then combining the result with "x", is the same as first combining "h" with "x" and then combining "g" with the result. Mathematically, a group action can be formally defined as follows: If "G" is a group and "X" is a set, a left group action of "G" on "X" is a function •: G × X → X such that: 1. For all "x" in X, e • x = x where "e" is the identity element in G. 2. For all "g", "h" in G and "x" in X, g • (h • x) = (gh) • x. This can similarly be defined for a right group action, but with the operations carried out in a different order. Group actions are an important concept used in various fields of mathematics, including group theory, geometry and algebra.

Freebase

  1. Group action

    In algebra and geometry, a group action is a description of symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set. In this case, the group is also called a permutation group or transformation group. A group action is an extension to the definition of a symmetry group in which every element of the group "acts" like a bijective transformation of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron. If G is a group and X is a set then a group action may be defined as a group homomorphism h 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 permutation of X assigned to:

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Numerology

  1. Chaldean Numerology

    The numerical value of group action in Chaldean Numerology is: 2

  2. Pythagorean Numerology

    The numerical value of group action in Pythagorean Numerology is: 4


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"group action." Definitions.net. STANDS4 LLC, 2023. Web. 2 Dec. 2023. <https://www.definitions.net/definition/group+action>.

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