On the Ellis Group of a Class of Distal Groups and a Problem by Milnes

Abstract

In (Acta Math Hangar 65(2):149–164, 1994), Milnes raised the problem that whether the Ellis group of a distal group is itself distal. In this paper, we give a positive answer to Milnes problem for the case of group extensions of Abelian groups. We also investigate the Ellis groups of a well-known family of distal compact right topological groups, namely the groups $$ E\left( {\mathbb{T}} \right)^{\infty } , E\left( {\mathbb{T}} \right)^{k} $$ and $$ E\left( {\mathbb{T}} \right)^{k - 1} \times {\mathbb{T}} $$, where $$ E\left( {\mathbb{T}} \right) $$ is the group of all endomorphisms on the unit circle $$ {\mathbb{T}} $$.