In terms of building our knowledge of Group Theory, we have two themes this week, both discussed in Section 1.3 of the lecture notes.
Subgroups
The first is the concept of a subgroup. Two things that algebraists like to do are find substructures of algebraic structures, that have similar basic properties, and to find ways of building larger structures from smaller ones. For example we can build up higher-dimensional spaces from the real line, but using multiple copies of it as coordinate axes. Here is a very short summary of the key points about subgroups, details are in the lecture notes and the short video which has a few examples.- A subset H of a group G is a subgroup of G if H is itself a group under the operation of G.
- What needs to be checked to verify this is that H is closed under the operation of G, that it contains the identity element of G, and that it contains the inverse in G of each of its own elements.
- Every group is a subgroup of itself and also has a subgroup consisting only of the identity element (this one is called the trivial subgroup). Occasionally these are the only subgroups, but usually there are some intermediate ones.
- If x is an element of the group G, then the smallest subgroup of G that contains the element x is called the cyclic subgroup of G generated by x, and we can describe exactly how to construct it from x. The cyclic subgroup generated by an element is a very important concept in group theory.
The concepts here are expressed in completely general and abstract terms. We construct definitions and arguments in some hypothetical abstract group, without knowing what the group is and with only the axioms at our disposal. This sort of setting is very standard in algebra, but it can be a bit disorienting at first. In this case a good strategy is to choose some example that you know well, think about what the abstract reasoning says about your example, and try to practise moving between the abstract and the more concrete settings.
Generating Sets
Here is a short summary of the key points, details are in the lecture notes.- A group is called cyclic if it contains an element a for which the cyclic subgroup generated by a is the whole group. In this situation a is called a generator of the group.
- As we did for single elements in the discussion about cyclic subgroups, we can think about the smallest subgroup of a group that is contains a particular pair of elements, or three particular elements. This is the subgroup generated by that set of elements.
- A subset S of elements in a group G is a generating set of G if every element of G can be written in terms of elements of S and their inverses.
Activity for Week 3 consists of the following steps.
- Make sure you are familiar with the content of Section 1.2 of lecture notes, on the definition of the group, and that you can relate the definition to the examples that we have looked at in Section 1.1.
- Come to the lectures on Thursday (on subgroups) and Friday (on generating sets).
Slides for both lectures: Lectures 5 and 6
If you cannot make it to the lectures, have a look at these video lectures from on the same topics (as usual, please ignore any details about assessment etc. that are not current for this year).
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Review the lecture notes for Section 1.3, on subgroups and generating sets
- The first of our two homework assignments will be posted here this week, with a deadline of October 14.