4
$\begingroup$

I have the following problem: "Let $G$ be a locally compact group, all of whose normal subgroups are contained in $Z(G)$. Prove that $G$ is unimodular."

My attempt at attacking the problem was to consider the homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ defined by \begin{equation*} \int_G f(yx)d_rx = \Delta(y)\int_Gf(x)d_rx \end{equation*} where $d_rx$ is a right Haar measure on $G$. Then $G$ will be unimodular if and only if $\Delta = 1$. However, I am having some difficulty showing that this holds. Any advice is appreciated.

  • 0
    Joel Cohen: A similar argument shows that $Z(G) \leq \ker \Delta$ so that certainly helps.2012-10-15

0 Answers 0