Short proofs of Hiramine' results on character values (Q1567134)

Let \(G\) and \(H\) be finite groups of order \(n\). A mapping \(f\colon G\to H\) is called a planar function of degree \(n\) if, for each \(\tau\in H\) and \(u\in G^\#\), there exists exactly one \(x\in G\) such that \(f(ux)f(x)^{-1}=\tau\). \textit{Y. Hiramine} [J. Algebra 152, No. 1, 135-145 (1992; Zbl 0769.20010)] proved the following result: If \(G\) and \(H\) are Abelian groups of order \(3p\geq 15\) with \(p\) a prime, then there exists no planar function from \(G\) into \(H\). To prove this he has established two results on character values. The author presents shorter proofs of these results.