Abstract

The Nottingham group was introduced as a pro-$p$ group by D. Johnson and his Ph.D student I. York. Before then it was known by number theorists as the group of normalized $F$ algebra automorphisms of the field of fractions of the formal power series ring over the field $F$ of prime characteristic. In other words, it is the group of formal power series with leading term $x$, under formal substitution. Since being introduced as a pro-$p$ group, it has been investigated by many group theorists, such as, A. Weiss, C. Leedham-Green, A. Shalev, R. Camina, Y. Barnea, B. Klopsch, I. Fesenko, M. Ershov, C. Griffin, P. Hegedus, M. du Sautoy and S. McKay. It gained interest as a pro-$p$ group after the result of R. Camina: The Nottingham group, over a field of characteristic $p$, contains an isomorphic copy of every finitely generated pro-$p$ group. Also, it has an important role in the theory of classification of just-infinite pro-$p$ groups, which are the simple objects in the category of pro-$p$ groups. The commutator structure of the Nottingham group is tight and well behaved. On the other hand, the $p$ th powers drop quickly, not yielding any structure. In my talk, after a survey about the Nottingham group and pro-$p$ groups in general, I will introduce a matrix (which was initially defined by K. Keating) to compute a certain type of commutator. It has a key role to prove a sharp bound for the distance of higher $p$ th powers of two elements of the Nottingham group, which was conjectured by K. Keating.