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Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$.

Here, $\varphi\times \eta$ is a character of $G$ defined by $(\varphi\times\eta)(h,k)=\varphi(h)\eta(k)$.

Thanks in advance.

  • 1
    Since $\varphi(h)$ is a sum of ${\rm deg}(\varphi)$ roots of 1 for all $h \in H$, and similarly for $\eta(k)$ for $k \in K$, we can only have $\varphi(h)\eta(k) = {\rm deg}(\varphi){\rm deg}(\eta)$ if $h$ and k are both mapped onto scalar matrices by the corresponding representations, and in that case $h \in Z(H)$, $k \in Z(K)$.2012-12-08

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