Normal subgroups in substitution groups of formal power series
Let \(R\) be a commutative ring with 1. This paper is concerned with the group \(N(R):=t+t^2Rt\) of normalized formal power series over \(R\) under substitution. If \(K\) is a finite field, \(N(K)\) is known as a Nottingham group that possesses a list of peculiar properties. The author studies the normal structure of \(N(R)\) in general for a commutative ring \(R\) with unity. If \(2\) is invertible in \(R\), the author proves that every normal subgroup of \(N(R)\) is closed. He gives even more precise information on the normal structure that looks especially nice if \(R=K\) is a field of characteristic \(p>2\). Put \(N_k(R):=t+t^{k+1}Rt\), \(k\in\mathbb{N}\). Then the following theorem is proved. Theorem 1.3. Let \(K\) be a field of characteristic \(p\neq 2\). Then every (abstract) closed normal subgroup of \(N(K)\) is closed. Moreover, for every non-trivial normal subgroup \(H\triangleleft N(K)\) there exists \(k\in\mathbb{N}\) such that one of the following holds (i) \(N_{k+1}(K)\subseteq HN_k(K)\) where \(k\not\equiv 1\pmod p\); (ii) \(N_{k+2}(K)\subseteq HN_k(K)\) where \(k\equiv 1\pmod p\). Further, a type of the congruence subgroup problem is studied, i.e., the relation between the groups \(\widehat{N(R)}\) and \(N(\widehat R)\). First the author shows that a subgroup of finite index of \(N(R)\) is open in the topology given by \(N_k(R)\). Using this the author deduces that \(\widehat{N(R)}=N(\widehat R)\) for a saturated ring \(R\), i.e., a ring \(R\) in which every Abelian subgroup of finite index contains an ideal of finite index. Finally, it is shown that for \(R=\mathbb{F}_pt\) the groups \(\widehat{N(R)}\) and \(N(\widehat R)\) are not isomorphic.
scientific article; zbMATH DE number 1475239