In which we wonder what a non-integer lecture is anyway.
In which we prove Lagrange’s theorem, and deduce many interesting results as a consequence.
In which we think about the link between equivalence relations and partitions, and meet cosets.
In which we think about cyclic groups, and renew an old friendship with equivalence relations.
In which we find that the alternating group is a group, and study subgroups in more detail.
In which we explore permutations in more detail.
In which we start to explore permutations.
In which we learn what a group is, and meet many examples.
Welcome to the course blog for the Oxford Prelims Groups and Group Actions course (Hilary and Trinity Terms 2015). I hope that this will be a useful resource to accompany the lectures, problems sheets and tutorials. Please check back after each lecture. You can easily find relevant posts using the categories on the right-hand side of the screen. In addition, I have a course page with some useful information, and you will also want to visit the official department course page (with problems sheets, for example).
The plan is that I’ll put up a new post just after each lecture. Each post will have a quick summary of the topics covered in that lecture, with suggestions for further reading, plus one or more problems to get you thinking about the topics that will be covered in the next lecture. Please do try these problems, as they will help you to get the most from the lectures (even if you don’t solve a problem, thinking about it will still be useful). There will also be exercises that you can try to check your understanding of that day’s lecture. Again, please do try these, as they will help you to work actively on your lecture notes and will get you thinking in the right way to tackle the problems sheets.
You are very welcome to leave comments (for me and for your fellow students) on each post. For example, you might have your own suggestions for good places to read about the topics, or you might have another way to look at one of the ideas, or you might have a really good example that illustrates some interesting aspect of the material, or you might have a question that you’d like to raise. I also encourage you to post to let me and others know how you and the people you’re working with have got on with the problems for the next lecture; please feel free to share ideas here (they don’t have to be complete solutions).
Of course, now I have to suggest something to think about before Lecture 1! So here it is. If you haven’t come across a definition of a group before, then you could do some quick reading, for example this very accessible NRICH article. How many examples of groups have you come across in the Oxford course so far? (Hint: you’ve met lots!) Think of as many as you can, it’s really helpful when learning group theory to have several examples of groups in mind.