A note on the Steinlein theorem (Q1280318)

The main result of the paper is as follows. If on the space \(X\) of the problem the periodic homomorphism \(\sigma\) has a simple period \(p\) and there exists a mapping \(g\: X\to\mathbb R^m\) such that \(g(x)\neq g(\sigma^kx)\) for any point \(x\in X\) for some \(k\), \(q\leq k\leq p-1\), then for every set of integers \(1\leq k_1,\dots,k_r\leq p-1\) a mapping \(f\: X\to \mathbb R^{m+r}\) exists such that \(f(x)\neq f(\sigma^{k_i}x)\) for all \(k_i\), \(i=1,\dots,r\) and all \(x\in X\).