# Group action

In mathematics, a **group action** is a relation between a group *G* and a set *X* in which the elements of *G* act as transformations (unary operations) on the set.

Formally, a group action is a map from the Cartesian product *G*×*X* to *X*,
written as or *x**g* or *x*^{g} satisfying the following properties (1_{G} being the neutral element of *G*):

*x*^{gh}= (*x*^{g})^{h}.

From these we deduce that , so that each group element acts as an invertible function on *X*, that is, as a permutation of *X*.

If we let *A*_{g} denote the permutation associated with action by the group element *g*, then the map from *G* to the symmetric group on *X* is a group homomorphism, and every group action arises in this way. We may speak of the action as a *permutation representation* of *G*. The kernel of the map *A* is also called the kernel of the action, and a **faithful action** is one with trivial kernel. Since we have

where *K* is the kernel of the action, there is no loss of generality in restricting consideration to faithful actions where convenient.

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## [edit] Examples

- Any group acts on any set by the
*trivial action*in which*x*^{g}=*x*. - The symmetric group
*S*_{X}acts of*X*by permuting elements in the natural way. - The automorphism group of an algebraic structure acts on the structure.
- A group acts on itself by right translation.
- A group acts on itself by conjugation.

## [edit] Stabilisers

The **stabiliser** of an element *x* of *X* is the subset of *G* which fixes *x*:

The stabiliser is a subgroup of *G*.

## [edit] Orbits

The **orbit** of any *x* in *X* is the subset of *X* which can be "reached" from *x* by the action of *G*:

The orbits partition the set *X*: they are the equivalence classes for the relation defined by

If *x* and *y* are in the same orbit, their stabilisers are conjugate.

The elements of the orbit of *x* are in one-to-one correspondence with the right cosets of the stabiliser of *x* by

Hence the order of the orbit is equal to the index of the stabiliser. If *G* is finite, the order of the orbit is a factor of the order of *G*.

A **fixed point** of an action is just an element *x* of *X* such that *x*^{g} = *x* for all *g* in *G*: that is, such that *O**r**b*(*x*) = {*x*}.

### [edit] Examples

- In the trivial action, every point is a fixed point and the orbits are all singletons.
- Let π be a permutation in the usual action of
*S*_{n}on . The cyclic subgroup generated by π acts on*X*and the orbits are the cycles of π. - If
*G*acts on itself by conjugation, then the orbits are the conjugacy classes and the fixed points are the elements of the centre.

## [edit] Transitivity

An action is **transitive** or **1-transitive** if for any *x* and *y* in *X* there exists a *g* in *G* such that *y* = *x*^{g}. Equivalently, the action is transitive if it has only one orbit.

More generally an action is **k****-transitive** for some fixed number *k* if any *k*-tuple of distinct elements of *X* can be mapped to any other *k*-tuple of distinct elements by some group element.

An action is **primitive** if there is no non-trivial partition of the set *X* which is preserved by the group action. Since the orbits form a partition preserved by this group action, primitive implies transitive. Further, 2-transitive implies primitive.

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