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Concept
Group action
Summary
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of GL(n, K), the group of the invertible matrices of dimension n over a field K.
The symmetric group S_n acts on any set with n elements by permuting the elements of the set. Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality.
If G is a group with identity element e, and X is a set, then a (left) group action α of G on X is a function
that satisfies the following two axioms:
{|
|Identity:
|
|-
|Compatibility:
|
|}
(with α(g, x) often shortened to gx or g ⋅ x when the action being considered is clear from context):
{|
|Identity:
|
|-
|Compatibility:
|
|}
for all g and h in G and all x in X.
The group G is said to act on X (from the left). A set X together with an action of G is called a (left) G-set.
From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to g ⋅ x is a bijection, with inverse bijection the corresponding map for g−1. Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself.
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