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Let  be a group and let  be a set. A left group action is a function
 such that:
-
 for all 
-
 for all
 and 
A right group action is a function
 such that:
-
 for all 
-
 for all
 and 
There is a correspondence between left actions and right actions, given by associating the right action  with the left action
 . In many (but not all) contexts, it is useful to identify right actions with their corresponding left actions, and speak only of left actions.
Special types of group actions
A left action is said to be effective, or faithful, if the function
 is the identity function on  only when  .
A left action is said to be transitive if, for every
 , there exists a group element  such that
 .
A left action is free if, for every  , the only element of  that stabilizes  is the identity; that is,
 implies  .
Faithful, transitive, and free right actions are defined similarly.
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