Let be a group, and let be a -set. The orbit of an element is the set , i.e., the set of conjugates of , or the set of elements in for which there exists for which .
For , the mapping is sometimes known as the orbital mapping defined by ; it is a homomorphism of the -set (with action on itself, by left translation) into ; the image of is the orbit of . We say that acts freely on if the orbital mapping defined by is injective, for all .
The set of orbits of is the quotient set of under the relation of conjugation. This set is denoted , or . (Sometimes the first notation is used when acts on the left, and the second, when acts on the right.)
Let be a set acting on from the right, and let be a normal subgroup of . Then acts on from the right, under the action , for . ( acts trivially on this set, so .) Consider the canonical mapping . The inverse images of elements of under are the orbits of under action of ; thus on passing to the quotient, defines an isomorphism from to .
Suppose and are groups, and acts on on the left, and on the right; suppose furthermore that the operations of and commute, i.e., for all , , , Let be the opposite group of ; then the actions of and on define a left action of on . The set is denoted . Since and are normal subgroups of , by the previous paragraph, the -sets , , are isomorphic and identitfied with each other.
Let be a group, and a subgroup of ; let it act on from the right. Then the set is the set of left cosets mod .
If is a group and are subgroups of , then the set is called the set of double cosets mod and .