Let $H_\lambda=\lambda\mathbb{Z}$ for $\lambda\in \mathbb{R}$ be a subgroup of the 1-dimensional translation group $T$. Consider then the factor group $K_{\lambda}=T/H_{\lambda}$ with representation $U_{\lambda}(x)$ for $x\in K_{\lambda}$. Then are the following two points correct:
1) $\exists$ a mapping $x'\in T \rightarrow x=x'H_\lambda \in K_\lambda$, which is in general many-to-one. $U_\lambda$ is also a representation of $T$ but a degenerate one.
2) Can $\lambda$ be interpreted as the wavelength of a wave and $U_{\lambda}(x)=e^{-i\frac{2\pi}{\lambda} x}$?