The length and depth of a group
Algebra and Geometry Seminar
16th May 2018, 2:30 pm – 3:30 pm
Howard House, 4th Floor Seminar Room
The length of a finite group G is defined to be the maximal length of an unrefinable chain of subgroups from G to 1; this notion has been the subject of numerous papers dating back to the 1960s, especially in the context of simple groups. In recent joint work with Martin Liebeck and Aner Shalev, we study a related concept, which we call the depth of G. This is the minimal length of an unrefinable chain of subgroups from G to 1 and it is interesting to compare these two parameters. In this talk, I will focus on the depth of simple groups. In particular, I will discuss our classification of the simple groups of minimal depth, and I will explain the somewhat surprising fact that alternating groups have bounded depth. Time permitting, I will conclude by highlighting some results for arbitrary finite groups and I will briefly mention some recent work on analogous notions of length and depth for algebraic groups and Lie groups.