Normal subgroups of symmetric groups, with an application to definability
by
Nathan Bowler
University of Cambridge
Coauthors: Thomas Forster

We will show that symmetric groups on infinite sets have no nontrivial normal subgroups of small index. Since the assumptions we'll need are weak (in particular we won't use AC), we will be able to apply the result in the context of type theories such as TST, TNT and NF, limiting the sizes of orbits under the induced action of the symmetric group of the universe.

In particular, we'll derive a wellfoundedness result for small symmetric sets in these contexts. We'll also present a complementary result limiting the sizes of wellfounded sets in NF.

Date received: March 20, 2008