As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity.
Given a locally compact abelian group $G,$ a closed subgroup $H,$ and a character $\chi: H \rightarrow S^1,$ I need to construct an extension of $\chi$ to all of $G.$ Any extension will do, but it's not clear to me that one should be able to construct an extension. For example, we can't just define an extension to be 1 outside of $H,$ since elements outside of $H$ can multiply to something within $H.$
Can anyone help me out? Thanks!