Generalized Hadamard matrices
Let \(\pi\) be a projective plane of order \(n\) such that \(\pi\) admits a group \(G\) of \(n\) elations each with axis \(L\) and center \(P\in L\). The existence of the elation group \(G\) is equivalent to the existence of a matrix \(H=(g_{ij})\) of order \(n\) with entries \(g_{ij}\) from \(G\) such that whenever \(m\not= k\) the set \(\{g_{mi}g_{ki}^{-1}\mid i=1,\dots,n\}\) contains each element of \(G\) once and only once. The author adds two conditions: (ii) \(g_{1i}=g_{i1}=1\) for each \(i\) and (iii) \(g_{ij}=g_{ji}^{-1}\) for all \(i,j\), and proves that \(G\) is a \(2\)-group and \(n=2^r\), and that if \(G\) is abelian or \(n\) is non-square \(G\) is elementary abelian. The proof uses character theory, in particular the Frobenius reciprocity theorem, quite ingeniously.
scientific article; zbMATH DE number 1051431