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Remarks on Hadamard groups. II. (English) Zbl 0892.20016
[For part I see Kyushu J. Math. 50, No. 1, 83-91 (1996; Zbl 0889.05033).]
Let \(G\) be a group of order \(8n\) containing a central involution \(e^*\), and let \(H\) be a subgroup containing \(e^*\). Denote by \(A(H)\) the set of \(x\in H\) such that \(x^2=e^*\). It is proved that if \(| A(G)|>7| G|/12\) then \(| A(G)|=3| G|/4\) or \(5| G|/8\). A characterization of the groups achieving the above equalities is given.

MSC:
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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