When we have a group acting on a set, we are often interested in how the group acts on a particular element. One natural way to do this is to look at the set of all the elements that different group actions will take the starting element to. This is called the orbit.

Definition

Let \(X\) be a set, with element \(x \in X\), and let \(G\) be a group acting on \(X\). Then the orbit of \(x\) is \(Gx = \{gx : g \in G\}\).

Properties

The set \(X\) is partitioned by group orbits—each element defines its own orbit, and two orbits containing the same element must be the same, because every action has an inverse action. This gives the fact that cosets partition a group as the special case where \(X\) is a subgroup of \(G\).

See Also

• The orbit-stabiliser theorem.