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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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