## Group Actions

Let be a set and let be a group who elements act on the set A left group action is a function such that:

1. for all where is the identity element in 2. for all and Right actions are similarly defined.

From these two axioms, it follows that for every the function which maps to is a bijective map from to Therefore, one may alternatively and equivalently define a group action of on as a group homomorphism from to  the set of all bijective maps from to If a group action is given, we also say that acts on the set or is a - set.

• Every group acts on itself in two natural ways: for all or for all • The symmetric group and its subgroups act on the set by permuting its elements.

• The symmetry group of a polyhedron acts on the set of vertices of that polyhedron.

• The symmetry group of any geometrical object acts on the set of points of that object.

• The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).

• The Matrix groups and act on • The Galois group and every subgroup of a field extension acts on the bigger field • The additive group of the real numbers acts on the phase space of systems in classical mechanics (and in more general dynamical systems): if and is in the phase space, then describes a state of the system, and is defined to be the state of the system seconds later if is positive or seconds ago if is negative.

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