Group orbit

by Adele Lopez Jun 27 2016 updated Jun 28 2016

When we have a group acting on a set, we are often interested in how the group acts on a particular [group_element 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.


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\}$~$.


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$~$.

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