Groups and Group Actions: Lecture 1

In which we learn what a group is, and meet many examples.

Here’s a quick summary of what we did in the lecture.  The links are to places where you can read more (sometimes about the maths, sometimes about the mathematicians).  Please feel free to use the comments facility!

  • Definition of a binary operation on a set.
  • Definition of an associative binary operation.
  • Definition of an identity element for a binary operation.
  • Proposition 1: Let \ast be a binary operation on a non-empty set S.  If there is an identity e \in S, then it is unique.  This was a straightforward proof: we showed that if e_1 and e_2 are both identity elements, then e_1 = e_2.
  • Definition of an inverse of an element (for a given binary operation).
  • Proposition 2: Let \ast be an associative binary operation on a set S, with an identity e.  Take a \in S.  If a has an inverse, then the inverse is unique.  Another straightforward argument showed that if b and b' are both inverses for a, then b = b'.
  • Definition of what it means for a subset to be closed under a binary operation.
  • Definition of a group.
  • Proposition 3: Let G be a group, let g, g_1, g_2, g_3 \in G, let m, n \in \mathbb{Z}.  Then
    1. (g_1 g_2)^{-1} = g_2^{-1} g_1^{-1};
    2. (g^n)^{-1} = g^{-n};
    3. g^m g^n = g^{m+n};
    4. (g^m)^n = g^{mn};
    5. if g_1 g_2 = g_1 g_3 then g_2 = g_3;
    6. if g_1 g_2 = g_3 g_2 then g_1 = g_3.

    The proof is an exercise in using the group axioms.

  • Definition of an Abelian group.

Understanding today’s lecture

We saw lots of examples and lots of non-examples of the various things defined in lectures.  It is worth checking that you can show carefully from the definitions that examples and non-examples really are examples and non-examples (as appropriate).  Can you come up with an additional example and non-example for each definition, perhaps drawing on things you’ve come across in other courses so far in Oxford?

You should have a go at proving the parts of Proposition 3 — I recommend proving enough of the parts that you feel confident that you could prove the rest without difficulty (if necessary, prove all six parts, it’s good practice!).

There was a question about the definition of a^n if n is negative.  Well, we define it to be a^{-1} multiplied by itself -n times, in the same sort of way as we define a^n.  And then Proposition 3 links this with the inverse of a^{-n}.

Further reading

You’ll want to look at Richard Earl’s printed notes from when he lectured the course last year.  My lectures are to a large extent based on his notes.

The internet is full of introductions to group theory, such as this very accessible article on NRICH, which motivates the ideas in a nice way and which links with things we’ll see in the next couple of lectures.  It turns out I sort of wrote one myself on this blog a while back, looking specifically at some modular arithmetic (which we’ll meet later in the course but which you could read up on now in the blog post).  Obviously Wikipedia has something to say on the subject, and if you want to know a little about the history of group theory then you could look here and here (this latter is based on a lecture given by Peter Neumann).

By the end of the course, we’ll have covered many although not all of the ideas in this video.  You can use it to check your grasp of the course as we progress: how many of the jokes do you understand?

Preparation for Lecture 2

Mattresses on beds need turning every so often, so that they wear evenly.  What are the symmetries of a mattress?  I can rotate it by certain amounts about certain axes (and still have a mattress that fits on my bed).  Check that the symmetries of a mattress form a group (the binary operation here is doing one symmetry then doing another).

What are the symmetries of an equilateral triangle?  They form a group, what is the size of the group?

What are the symmetries of a square?  They form a group, what is the size of the group?

Can you generalise to a regular n-gon?

If I give you two groups (G, \ast_G) and (H, \ast_H) (note that I have allowed them to have different binary operations), how might we turn the set G \times H into a group?  That is, how might we define a binary operation on G \times H using the binary operations on G and H?


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