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Group Action

Basic definition

A group action is the association of each of a mathematical group's elements with a permutation of the elements of a set.

Detailed definition

A group G is said to act on a set X when there is a map ϕ:G×X->X such that the following conditions hold for all elements x element X. 1.ϕ(e, x) = x where e is the identity element of G. 2.ϕ(g, ϕ(h, x)) = ϕ(g h, x) for all g, h element G. In this case, G is called a transformation group, X is a called a G-set, and ϕ is called the group action. In a group action, a group permutes the elements of X. The identity does nothing, while a composition of actions corresponds to the action of the composition. For example, as illustrated above, the symmetric group S_10 acts on the digits 0 to 9 by permutations.

Related terms

action | effective action | free action | Galois group | group | group block | group orbit | group representation | isotropy group | Lie group quotient space | matrix group | primitive group action | proper group action | topological group | transitive

Educational grade level

college level

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