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Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\...

Update: I have found reference to this problem. It is known as "the Rédei-de Bruijn-Schönberg theorem", which is proved in the following papers: N. G. de Bruijn: On the factorization of cyclic ...