Groups and Group Actions: Lecture 12

In which we meet group actions.

• Definition of a left action of a group on a set.
• Definition of a right action of a group on a set.
• Definition of the orbit and stabiliser of an element of a set under an action of a group.
• Proposition 56: The orbits of an action partition the set.  We defined an equivalence relation whose equivalence classes are precisely the orbits, and were then done by Theorem 27.
• Definition of what it means for a group to act transitively on a set.
• Proposition 57: Let  be a group acting on a set .  Take .  Then  is a subgroup of .  This was a quick check using the definition of the stabiliser and of the group action.

Understanding today’s lecture

We saw a few examples of group actions, and only checked the conditions for some of them.  It would be a great idea to check carefully that the others really are group actions, to get a feel for the definition.

For each action we saw in the lecture, pick an element of the set and find its orbit and stabiliser.

Can you give any more examples of actions?  Can you give an example of a group that acts on a set but not transitively?

Further reading

This blog post about group actions by Tim Gowers is well worth a read.

There are lots more examples of group actions on Wikipedia.

Perhaps you’re wondering why we can concentrate on left actions and not worry about right actions?  Have a think for yourself, and then there are lots of sources of help online.

Here is a helpful illustration of the action of a matrix on a vector.

Preparation for Lecture 13

Let  be a finite group acting on a set .  Take .  What can you say about the connection between the orbit and stabiliser of ?  (If you’ve played with some examples above then you’ll have some at hand to explore here.)