Let M(X) denote the group of all invertible maps X->X and let G be any group. A homomorphism theta:G->M(X) is called an action of G on X. Therefore, theta satisfies

1. For each g in G, theta(g) is a map X->X:x|->theta(g)x.

2. theta(gh)x=theta(g)(theta(h)x).

3. theta(e)x=x, where e is the group identity in G.

4. theta(g^(-1))x=theta(g)^(-1)x.

See also

Cascade, Flow, Group Action, Semidirect Product, Semiflow

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Cite this as:

Weisstein, Eric W. "Action." From MathWorld--A Wolfram Web Resource.

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