MA3343 Groups. Week 8: Group Actions

Welcome to our activities for Week 8 (the week of November 16).

Our theme this week is the concept of a group action. This is not a new instalment of theoretical content so much as a viewpoint that considers groups and their elements in terms of what they "do to other things" rather than as self-contained objects. The idea is that groups "act" on sets by moving the elements of the set around. We have seen several examples of this already - for example symmetries of polygons permute the vertices, permutations shuffle elements of a specified set, (invertible) matrices act as linear transformations of vector spaces, they permute the elements of the vector space. The language of group actions describes these ideas in a precise way and also allows us to make deductions, mostly using Lagrange's Theorem, about connections between the order and structure of a group and the ways in which it can permute the elements of a set. The most important words in the vocabulary of this language are action, orbit and stabilizer. After this week, you should be able to say what each of these words means. As usual we have two videos. The first one gives a few examples, they are familiar to us already but here they are described in the language and context of group actions.



Accompanying slides are here.

The second one formalizes the definition of a group action.



Accompanying slides are here, and the annotated version is here

Lecture Notes

This week's content is in Section 3.1 and Section 3.2 of the lecture notes, which as usual contain some extra detail and alternative examples to the ones in the videos.

Feedback on Week 8 challenge



The challenge for Week 8 had two elements - to show that the set in question is a group under matrix multiplication, and to describe the orbit of its action on the Cartesian plane. Most people who responded gave a complete answer, but many responses were virtually identical and possibly drawn from some source, which should have been cited if so.
The first part was answered very well on the whole, suggesting that students have a strong grasp of what is required to show that something is a group or subgroup, a very important skill in group theory. The most common omission on the first part was in showing that the set is closed under matrix multiplication. After showing that the product of two elements of G has the required pattern of elements, a few people neglected to confirm that the determinant of the product is 1.
The second part was also answered well, with most people showing that multplying a vector on the left by an element of G does not change its length. The most satisfying explanations of how this group acts on the plane in my opinion were given by authors who noted that a pair of real numbers a and b with a2+b2=1 are the cosine and sine of some θ, and that the matrices in G are exactly the rotations of the plane about the origin. The orbits of the action are the circles centred at (0,0).
Here is one excellent response.