group action
Let be a group and let be a set. A left group action is a function such that:
-
1.
for all
-
2.
for all and
A right group action is a function such that:
-
1.
for all
-
2.
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.
Title | group action |
Canonical name | GroupAction |
Date of creation | 2013-03-22 12:12:17 |
Last modified on | 2013-03-22 12:12:17 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 10 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 16W22 |
Classification | msc 20M30 |
Related topic | Group |
Defines | effective |
Defines | effective group action |
Defines | faithful |
Defines | faithful group action |
Defines | transitive |
Defines | transitive group action |
Defines | left action |
Defines | right action |
Defines | faithfully |
Defines | action |
Defines | act on |
Defines | acts on |