On identities in connected topological groups

Abstract

In 1957, Nemytskii proved the following fact: if in a locally compact or in an Abelian connected group there is a neighborhood of the identity in which some identity holds, then it holds in the entire group. The following question was also posed there: Let G be a connected topological group. In some neighborhood of the identity of the group G the identity $x^3=1$ holds. Is it true that then the identity $x^3=1$ holds in the entire group $G$? The same question is posed for the identity $gx^2 = x^2g$, where $g$ is a fixed element of the group. Platonov formulated the following generalized formulation of the Mytselsky problem: for a topological connected group, is it true that if the identity holds in a neighborhood of the identity, then the identity holds everywhere? In this paper, a negative answer to Platonov's question is given, the following theorem is proven: if $n > 10^{10}$ is odd, then there exists a connected topological group in which the identity $x^n=1$ holds in some neighborhood of unity, but not in the entire group.