Group orbit

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 $$G$$.