Finite groups act on sets by permuting their elements. For instance, each group acts on the set of its own elements by multiplication. This action is faithful because every element in the group acts nontrivially, i.e. multiplication by it is not boring. This faithful action actually gives an injective homomorphism G to S_n, where n is the size of G.
I will discuss the following natural question: Given a group G of size n, what is the SMALLEST m such that G injects into S_m? For instance, Z/6Z injects into S_5, but not any smaller S_n. What does this data tell you?
I will discuss the implications of this to the Higman PORC conjecture and to the classification of finite p-groups.
I will assume a little bit of group theory - I will try to explain the basics fast, but unless you've had group theory it'll be tricky. If you've had group theory, it's very elementary.
I gave this talk to UMS 2 years ago. If you were there, you haven't proven the Higman PORC conjecture yet, so you might as well come again.