Let be a group and be a set. Then is called a left -set if there exists a map such that

for all and all . This is commonly written , so the above relation becomes

The map is called a left -action on the set .

Right -sets and right -actions are defined analogously except elements of are multiplied by elements of to the right instead of to the left. Left -sets and right -sets are both called -sets for simplicity.

A -set is an example of a group set, where is the group in question.