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.
Slides for this week’s lectures.
Lecture Notes: This week’s content is in Section 3.2 and Section 3.3 of the lecture notes, which as usual contain some extra detail and some alternative examples to the ones in the lectures.
Video versions of the lectures are below.