In which we meet Cayley’s Theorem and reach the end of this particular adventure, but catch a glimpse of far-off lands still to be explored.
- Theorem 68: Let be a group, let be a set.
- Given a left action of on , there is an associated homomorphism .
- Given a homomorphism , there is an associated left action of on .
- These correspondences are inverses of one another.
We proved this by just writing down suitable homomorphisms/actions and checking that they work, there were no major ideas involved.
- Corollary 69 (Cayley‘s Theorem): Let be a finite group. Then is isomorphic to a subgroup of for some . The key here is that acts on itself by left multiplication, and using Theorem 68 this leads to a homomorphism from to .
Understanding today’s lecture
Cayley’s Theorem shows that if is a group of order then is isomorphic to a subgroup of . You might like to revisit Sheet 7 Q5 to see how this relates to what you found there (where you were asked for the smallest such that is isomorphic to a subgroup of ).
What is the group of rotational symmetries of a tetrahedron?
There’s a Wikipedia page about Cayley’s theorem.
Here’s Tim Gowers writing about a couple of past Cambridge exam questions, including one with a rather nice application of the theory of group actions.
Here’s Terry Tao on a generalisation of Cayley’s theorem (this comes with a health warning: the post is not written for first-year undergraduates!).
There are loads of ideas that build on group theory. Coming up later in the Oxford course you’ll find
and that’s without the fourth year courses, and without the many courses that use group theory along the way. Note that group theory is hugely important in other subjects such as modern physics, it’s not just for the pure mathematicians. See for example this video about group theory and physics, or this Plus article (with podcast) about group theory and viruses, or this Plus article about group theory and chemistry, or these lecture notes about group theory and physics. Or just try a library or Google!