Abstract

The question which I hope to address in this lecture is: Why care about cycles in permutation groups?  It has engaged mathematicians for around 150 years, going back to Jordan’s seminal result that the finite primitive permutation groups containing a prime length cycle with at least three fixed points are the giants, the alternating group and symmetric group. From the 1970’s, this and other old results offered a means of identifying these giants among primitive groups computationally, using random selections to find such elements. The computational application raised a further question: Just how easy is it to find, or construct, one of these Jordan cycles?  I’ll trace this story up to 2021, when with Stephen Glasby and Bill Unger we found a (to us surprising) answer.   If there is time I’ll draw a parallel with stingray matrices in finite classical groups.