MA3343 Groups, Week 2 21/22: The definition of a group

Welcome to our activities for Week 2 (the week of September 13). In terms of building our knowledge of Group Theory, there are two goals this week.

To add groups of symmetries and groups of permutations to our collection of examples from Week 1. We have already looked at the symmetries of friezes, but have not interpreted them in terms of binary operations with identity elements, pairs of inverse elements etc. That gap can hopefully be filled by this Thursday's lecture on groups of symmetries and groups of permutations, and by the lecture notes.
To become familiar with the axiomatic definition of a group, and to relate each of the examples that we have encountered so far to that. A good exercise for getting a sense of what this definition is for, is to use it to decide whether a given algebraic structure is a group or not. There will be some tasks like this on the first homework sheet, and one in our second weekly challenge.

Activity for Week 2 consists of the following steps.

Have a look back at the examples from last week's lectures, and at Section 1.1 of the lecture notes.
Come to the lectures on Thursday and Friday, September 16 and 17.
Slides for Thursday's lecture: before and after
Slides for Friday's lecture: before and after
If you cannot make it to the lectures, have a look at these videos from last year (as usual, please ignore any details that are particular to last year).
Have a go at this week's challenge, and submit your response via the Blackboard assignmnet, by the deadline of 5pm on Monday September 27.
By the end of this week, our position in the lecture notes will be at the end of Section 1.2. There are some examples in the notes that are not discussed in the lectures, so have a look at them even if you have your own notes from the lectures. The relevant section of the lecture notes are linked here, including Section 1.3 which presents some more important concepts in group theory. We will have a look at those in next week's lectures.
Section 1.1 of lecture notes: Some examples
Section 1.2 of lecture notes: The axioms of a group
Section 1.3 of lecture notes: Subgroups and generating sets