# A generalization of the character group

Let $$G$$ be a group. We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$

where $$Tor(\mathbb{T})$$ is the group of torsion elements of the unit circle $$\mathbb{T}$$.

Then $$\tilde{G}$$ is a group with the obvious operation.Furthermore, if $$G$$ is a topological group, we equip $$\tilde{G}$$ with the compact open topology.

Assume that $$G$$ is a locally compact abelian group. Is true to say $$\tilde{G}$$ is a locally compact abelian topological group, too? Under which condition do we have the duality $$\tilde{\tilde{G}}\sim G$$

Is there a notion or terminology for this $$\tilde{G}$$?