Consider the oriented cobordism group of $BG$ with $U(1)$ coefficient $\Omega_{SO}^d(BG,U(1))$ with a finite discrete group $G$.
question: Explain or show that the Pontryagin-dual of the torsion subgroup of oriented bordism group $\Omega_{SO,d}(BG)$ is $$\frac{\text{Hom}(\Omega_{SO,d}(BG),U(1))}{ \text{im}(f)}.\;\;\;\; (?)$$
where $f$ is a map from $f: \text{Hom}(\Omega_{SO,d}(G),\mathbb{R}) \to \text{Hom}(\Omega_{SO,d}(G),U(1))$. The image of the $f$ map is composed by elements of $\text{Hom}(\Omega_{SO,d}(BG),U(1))$ that vanish on the torsion subgroup of bordism group $\Omega_{SO,d}(BG)$.