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

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.

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college level