About a structure of exponential monomials on some locally compact abelian groups

We describe the structure of some class of exponential monomials on some locally compact abelian groups. The main result of the paper is the next theorem. Let $\tilde{G}$ and $G$ be locally compact abelian groups, $\alpha : \tilde{G}\to G$ be a continuous surjective homomorphism and $H$ be a kernel of $\alpha$. If $\alpha$ is a an open maps from $\tilde{G}$ to $G$ then any exponential monomial $\Phi(t)$ on the group $\tilde{G}$, which satisfy the condition $\Phi(t+h)=\Phi(t)\forall h\in H, t\in \tilde{G}$, can be presented in the form $\Phi(t)=f(\alpha(t))$ for some exponential monomial $f(x)$ on the group $G$.