*In which we explore groups of order and encounter Cauchy’s Theorem.*

- Lemma 62:
*Let be a prime, let be a group of order . Then is Abelian.*We saw last time that the centre of , , is non-trivial, so in this case it must have order (and so be the whole group, making Abelian) or . In the latter case, we saw that is cyclic and used this to deduce that in fact is Abelian. - Proposition 63:
*Let be a prime, let be a group of order . Then is isomorphic to or .*In the case that contains no element of order , we picked suitable and and used the fact that is Abelian to show that . - Theorem 64 (Cauchy‘s Theorem):
*Let be a finite group, let be a prime dividing . Then contains an element of order .*We considered the set , and the action on it by where defined by . By considering the sizes of the orbits, we found a non-trivial orbit of size 1 (not just ), and showed that this corresponded to an element of order .

### Understanding today’s lecture

Would the arguments in Lemma 62 and Proposition 63 work for groups of order ? Is every group of order Abelian?

What are the groups with exactly two conjugacy classes?

What goes wrong in our proof of Cauchy’s Theorem if is not prime? (We know that the result isn’t true in general, don’t we? Can you give a group of order 4 that contains no element of order 4?)

### Further reading

As I mentioned, there are several results known as Cauchy’s Theorem.

I mentioned the structure theorem for finite Abelian groups, which is included in the Algebra 2 (Rings and Modules) course.

There are other ways to prove Cauchy’s Theorem, here is a poster for your bedroom wall.

### Preparation for Lecture 15

On Sheet 6, there was a question about counting orbits of colourings of the edges of a triangle under the action of , using first two and then three colours. Can you generalise to colours?

What is the group of symmetries of the cube? We talked about the size of the group of rotational symmetries, what is the size of the group of *all* symmetries? What is the structure of the group?

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