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of order 8.

Group actions

In the above example of the symmetry group of a square, the permutations "describe" the movement of the vertices of the square induced by the group of symmetries. It is common to say that these group elements are "acting" on the set of vertices of the square. This idea can be made precise by formally defining a group action.

Let G be a group and M a nonempty set. An action of G on M is

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