The Nottingham group \(N={\mathcal N}(\mathbb{F}_p)\) is the group of formal power series \(\{t(1+a_1t+a_2t^2+\cdots):a_i\in\mathbb{F}_p\}\) under substitution. Here \(\mathbb{F}_p\) denotes the field with \(p\) elements. This is a pro-\(p\) group that can be generated by two elements. A useful general reference for the remarkable properties of \(N\) is the survey of R. Camina [in M. du Sautoy (ed.), et al., New horizons in pro-\(p\) groups. Boston, MA: Birkhäuser. Prog. Math. 184, 205-221 (2000; Zbl 0977.20020)].
The main result of the paper under review is that \(N\) is finitely presented as a pro-\(p\) group, for \(p>2\). This is a very important result, that represents a major progress in our knowledge of this elusive group.
Let \(L\) be the graded Lie ring associated to \(N\) with respect to the lower central series. This is a Lie algebra over \(\mathbb{F}_p\). The finite presentability of \(N\) would follow from that of \(L\). However, the reviewer has proved [J. Algebra 198, No. 1, 266-289 (1997; Zbl 0977.17016)] that a certain central extension of \(L\) is finitely presented, whereas \(L\) itself is not. The author of the paper under review starts indeed from (the positive part of) this result, but there is still a long way to go to get from the Lie algebra to the group. To do this, the author builds upon the subgroups of \(N\) studied in a previous paper of his [J. Algebra 275, No. 1, 419-449 (2004; Zbl 1062.20030)]; some subtle calculations are required.
The presentation for \(N\) found here involves a number of relations that depends on \(p\), as in the case of \(L\). Computational evidence suggests that five relations should suffice, but probably new ideas are needed to prove this.