Welcome to our activities for Week 9 (the week of November 23).
This week we continue with the topic of group actions.
We have two key items, one in each of the video lectures.
The first is the Orbit-Stabilizer Theorem, which has great importance in group theory and its applications. Its statement relates the index of the stabilizer of an element to the size of the associated orbit, but one of its most useful immediate consequences is that the number of elements in any orbit under the action of a finite group must be a divisor of the group order.
The details of the theorem and its proof are in the first of our two videos for Week 9.
The second major item this week is Cayley's Theorem, another famous theorem about finite groups. A slightly vague way to describe it is to say that every group of order n can be imagined as a subgroup of the symmetric group Sn. "Can be imagined" is not a mathematically precise term obviously, what it means is explained a bit in the video, we'll tidy it up a bit more later when we formalize the concept of isomorphism. Two groups are isomorphic to each other if they are structurally identical and differ only in how their elements are labelled.
This week's content is in Section 3.2 of the lecture notes, which as usual contain some extra detail and alternative examples to the ones in the videos.