Group action

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Definition[edit | edit source]

Classically a group action is a function where G is a group with identity i and X is a set such that:

In this way the group action preserves some qualities of the group. These requirements ensure that for every element g in G, acts as a permutation on X. Thus this can equivalently be formulated to say that is a homomorphism from G to the group of permutations on X.

From here the definition can be simplified and generalized to: A group action is a group homomorphism from a group G to an automorphism group of some object X.

The permutation group of a set is its automorphism group, but this definition allows groups to act on more than just sets. For example it is useful to think of a group acting on an abstract polytope, that is a homomorphism from the group to the group of automorphisms on the polytope.

This notion can be extended further. For example despite requiring that G being a group this definition does not use the inverse property and thus works perfectly for monoids as well. And furthermore the closure isn't used so G can be relaxed further to simply be a category. With the definition relaxed this much a group action is equivalent to the definition of a functor.

Transitivity[edit | edit source]

An important concept for polytopes is the concept of transitivity. A group action acts transitively iff for any two elements x and y in X there is a group element g such that .

For example the definition of isogonal is that the symmetry group of a polytope acts transitively on its vertices. That is that for any two vertices of the polytope there is a member of the group that maps between them.

Orbit[edit | edit source]

For a group action and an element , the orbit of x is the set of elements .

The group action acts transitively on all orbits, and thus the orbits partition a set.

Every element is in its own orbit and an element whose orbit consists entirely of itself is called a fixed point.

Faithful actions[edit | edit source]

A faithful action is an action with the additional stipulation that it is an injective homomorphism. This can also be stated as:

That is, if an element of G maps to the identity automorphism, it must be an identity in the group.

Free actions[edit | edit source]

A free action is an action such that:

That is, if an element of G fixes an element of X, it must be an identity in the group. This is a stronger condition than faithful and all free actions are also faithful.

Examples[edit | edit source]

  • For a group G its own operator is a an action acting on itself. As a result sometimes the group operator itself is called a "group action".
  • For any group G and object X there is a trivial group action, which always yields the identity automorphism.

External links[edit | edit source]