Group action

It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane.

It can be notationally convenient to curry the action α, so that, instead, one has a collection of transformations αg : X → X, with one transformation αg for each group element g ∈ G. The identity and compatibility relations then read and with ∘ being function composition.

The second axiom then states that the function composition is compatible with the group multiplication; they form a commutative diagram.

With the above understanding, it is very common to avoid writing α entirely, and to replace it with either a dot, or with nothing at all.

The difference between left and right actions is in the order in which a product gh acts on x.

Equivalently, the homomorphism from G to the group of bijections of X corresponding to the action is injective.

A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free).

For instance the abelian 2-group (Z / 2Z)n (of cardinality 2n) acts faithfully on a set of size 2n.

This is not always the case, for example the cyclic group Z / 2nZ cannot act faithfully on a set of size less than 2n.

In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size.

The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.

If X is acted upon simply transitively by a group G then it is called a principal homogeneous space for G or a G-torsor.

The action of the symmetric group of X is transitive, in fact n-transitive for any n up to the cardinality of X.

The action of the general linear group of a vector space V on the set V ∖ {0} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of v is at least 2).

The action of the orthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on the unit sphere.

[6] The action by deck transformations of the fundamental group of a locally simply connected space on a universal cover is wandering and free.

For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space G \ X.

If g acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero g-invariant submodules.

The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X.

In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset.

The kernel N of the homomorphism with the symmetric group, G → Sym(X), is given by the intersection of the stabilizers Gx for all x in X.

Let (H) denote the conjugacy class of H. Then the orbit O has type (H) if the stabilizer Gx of some/any x in O belongs to (H).

Therefore f induces a bijection between the set G / Gx of cosets for the stabilizer subgroup and the orbit G⋅x, which sends gGx ↦ g⋅x.

This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well).

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

Fixing a group G, the set of formal differences of finite G-sets forms a ring called the Burnside ring of G, where addition corresponds to disjoint union, and multiplication to Cartesian product.

Some example isomorphisms: With this notion of morphism, the collection of all G-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

For example, if we take the category of vector spaces, we obtain group representations in this fashion.

We can view a group G as a category with a single object in which every morphism is invertible.

[16] A morphism between G-sets is then a natural transformation between the group action functors.