Functional equations and topological N-groups

Abstract.

Let N be a topological nearring. A topological N-group is a topological group G, together with a continuous function from N x G to G, sending (a, x) to ax, such that (a + b) x = ax + bx and (ab) x = a (bx) for all $a,b \in N$ and $x \in G$. We determine all topological N-groups in the case where the topological nearring N is the unique, up to isomorphism, topological nearring whose additive group is the two dimensional Euclidean group $\Bbb {R}^2$ and which has an identity but is not zero symmetric, and the topological group is any one of the n-dimensional Euclidean groups $\Bbb {R}^n$. A key ingredient in all this is the solution of a functional equation involving two continuous selfmaps of $\Bbb {R}^n$.