Groups and Group Actions: Lecture 16

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 G be a group, let X be a set.
    1. Given a left action of G on X, there is an associated homomorphism \rho : G \to \mathrm{Sym}(X).
    2. Given a homomorphism \rho : G \to \mathrm{Sym}(X), there is an associated left action of G on X.
    3. 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 G be a finite group.  Then G is isomorphic to a subgroup of S_n for some n The key here is that G acts on itself by left multiplication, and using Theorem 68 this leads to a homomorphism from G to \mathrm{Sym}(G).

Understanding today’s lecture

Cayley’s Theorem shows that if G is a group of order n then G is isomorphic to a subgroup of S_n.  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 k such that G is isomorphic to a subgroup of S_k).

What is the group of rotational symmetries of a tetrahedron?

Further reading

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!).

Where next?

You might like to visit hedgehogmaths when revising, if you are someone who finds videos helpful.

There are loads of ideas that build on group theory.  Coming up later in the Oxford course you’ll find

Rings and Modules

Group Theory

Introduction to Representation Theory

Galois Theory

Topology and Groups

and that’s without the fourth year courses, and without the many courses that use group theory along the way.

On a less serious note, there is some interesting group theory associated with the Rubik’s cube.

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!


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