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The paper studies a generalized Hadamard matrix H = (g_{ij}) of order n with entries g_{ij} from a group G of order n. We assume that H satisfies: (i) For m \neq k, G = \{g_{mi} g_{ki}^{-1}\mid i = 1, \ldots , n\}; (ii) g_{1i} = g_{i1} = 1 for each i; (iii) g_{ij}^{-1} = g_{ji} for all i, j. Conditions (i) and (ii) occur whenever G is a(P, L) -transitivity for a projective plane of order n. Condition (iii) holds in the case that H affords a symmetric incidence matrix for the plane. The paper proves that G must be a 2-group and extends previous work to the case that n is a square. |
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