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On Hadamard property of a certain class of finite groups. (English) Zbl 0911.20008
A finite group $$G$$ of order $$8n$$ with a central involution $$e^*$$ is called a Hadamard group if $$G$$ contains a transversal $$D$$ with respect to $$\langle e^*\rangle$$ such that $$| D\cap Dr|=2n$$ for every element $$r$$ of $$G$$ outside $$\langle e^*\rangle$$. Such a transversal is called a Hadamard subset. The cyclic group of order 4 is considered as a Hadamard group. If $$r^2=e^*$$, then we have $$| D\cap Dr|=2n$$ for every transversal $$D$$ of $$G$$ with respect to $$\langle e^*\rangle$$. Let $$A(e^*)$$ be the set of elements $$r$$ of $$G$$ such that $$r^2=e^*$$ and $$a(e^*)=| A(e^*)|$$. Then if $$a(e^*)$$ is large enough compared with $$| G|$$ we may expect $$G$$ to be Hadamard. In the present paper one assumes that $$a(e^*)\geq 4n$$ and we investigate the Hadamard property of $$G$$ (of order $$8n$$). Let $$T(G)$$ be the sum of the irreducible characters of $$G$$. Then $$T(G)\leq a(e^*)$$, by the Frobenius-Schur formula on the number of involutions. This means that $$2T(G)\geq| G|$$, and such groups were classified by the reviewer and K. G. Nekrasov [see Ya. G. Berkovich and E. M. Zhmud, Characters of finite groups, Part 1. Transl. Math. Monogr. 172, Am. Math. Soc., Providence (1998), Chapter 11]. The authors divide their groups into three classes: Class I consists of groups such that $$a(e^*)>4n$$, class II consists of groups $$G$$ such that $$a(e^*)=4n$$ and $$G$$ contains a nonreal element, and class III consists of groups $$G$$ such that $$a(e^*)=4n$$ and all elements of $$G$$ are real. For groups of classes I and II we have that $$2T(G)>| G|$$. Note that generalized quaternion groups are members of class I (it is known that these groups are Hadamard; semidihedral and dihedral groups are not Hadamard).
The main result of this paper is the following Proposition 3. Any 2-group of order $$8n$$ with $$a(e^*)\geq 4n$$ is Hadamard.
Note that there exists a non-Hadamard group among groups of class II. Namely, class II contains the series of groups $$X(n)$$ presented by $$X(n)=\langle r,s\mid r^{2n}=s^4=1$$, $$s^{-1}rs=r^{-1}\rangle$$.
Proposition 4. If $$X(n)$$ is Hadamard, then $$n$$ is a sum of two squares. In particular, $$X(3)$$ is not Hadamard.
The class of Hadamard groups is very large. Indeed, as Ito showed, for every 2-group $$P$$ there exists a Hadamard 2-group $$G$$ such that $$P$$ is isomorphic to a subgroup of $$G$$. All known groups of class I are Hadamard. Class I contains the series of groups $$G(n)=\langle r,s\mid r^{2n}=s^2=e^*$$, $$s^{-1}rs=r^{-1}\rangle$$. These groups were investigted in the following papers: A. Baliga and K. J. Horadam [Australas. J. Comb. 11, 123-134 (1995; Zbl 0838.05017)] and D. L. Flannery [J. Algebra 192, No. 2, 749-779 (1997; Zbl 0889.05032)]. There are many open questions on Hadamard groups (for example, abelian Hadamard groups are not classified).
##### MSC:
 20C15 Ordinary representations and characters 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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##### References:
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