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On Hadamard groups of order 72. (English) Zbl 0869.05017
Let \(G\) be a group of order \(8m\) with a central involution \(e^*\), and denote the group of order 2 generated by \(e^*\) by \(N\). What the authors call an Hadamard subset in \(G\) is—in more standard terminology—just a relative difference set with parameters \((4m,2,4m,2m)\) in \(G\), relative to the forbidden subgroup \(N\); for background on relative difference sets, see A. Pott [“A survey on relative difference sets”, Group, difference sets, and the monster (eds. K. T. Arasu et al.), Berlin: Walter de Gruyter, Ohio State Univ. Math. Res. Inst. Publ. 4, 195-232 (1996; Zbl 0847.05018)]. If \(G\) admits such an Hadamard subset, it is called an Hadamard group. The authors determine all Hadamard groups of order 72 (up to equivalence) and study the corresponding Hadamard matrices of order 36.

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures