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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.)