On identities in connected topological groups

TL;DR

This paper investigates whether local identities near the identity element in connected topological groups imply global identities, providing a negative answer for certain large odd exponents by constructing counterexamples.

Contribution

The paper demonstrates that for sufficiently large odd integers, local identities do not necessarily extend globally in connected topological groups.

Findings

Counterexamples exist for large odd n where x^n=1 holds locally but not globally

The result applies to n > 10^{10}, showing the limits of local-to-global identity extension

Provides a negative answer to a generalized form of Mytselsky's problem

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.