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 \(G\).

See Also

Parents:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.