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If $Z$ is a subgroup of the center of G and $|G:Z|=m$, then $\chi^*(g)=m\chi(g)$ if $g\in Z$ where $\chi$ is a character of $Z$ and $\chi^*$ is the induced character of $G$.

Let $\phi$ be the representation of $Z$, and $\Phi$ be the induced representation. I am having a hard time of finding the relation between the trace of $\Phi$ and when it is restricted to $\phi$. What can we say about the traces? From the question it seems that the traces "differ" by $m$ and the idea of Schur's lemma will be helpful in this case.

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The induced character $\chi*$ is given by $$\chi^*(g)=\frac{1}{|Z|}\sum_{x\in G}\chi(xgx^{-1}),$$ where the summand is interpreted as $0$ if $xgx^{-1}\not\in Z$. But then, as $Z\subset Z(G)$, $xgx^{-1}=g$ for all $x\in G$, and so this becomes $$\chi^*(g)=\frac{1}{|Z|}\sum_{x\in G}\chi(g)=\frac{1}{|Z|}|G|\chi(g)=\frac{1}{|Z|}|Z|m\chi(G)=m\chi(g).$$

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