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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).
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.)
Full Text: DOI
[1] Baliga, A.; Horadam, K.J., Cocyclic Hadamard matrices overZ_t√óZ22, Australas. J. comb., 11, 123-134, (1995) · Zbl 0838.05017
[2] Berkovich, Y.; Zhmud’, E., Characters of finite groups, part I, translations of mathematical monographs, (1997), Amer. Math. Soc
[3] Cho, J.R.; Ito, N.; Kim, P.S.; Sim, H.S., Hadamard 2-groups with arbitrarily large derived length, Australas. J. comb., 16, 83-86, (1997) · Zbl 0892.20015
[4] Feit, W., Characters of finite groups, (1967), Benjamin New York/Amsterdam · Zbl 0166.29002
[5] Flannery, D.L., Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. algebra, 192, 749-779, (1997) · Zbl 0889.05032
[6] Ito, N., Some results on Hadamard groups, (), 149-155 · Zbl 0864.05021
[7] Ito, N., Note on Hadamard groups and difference sets, Australas. J. comb., 11, 135-138, (1995) · Zbl 0826.05010
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[10] Ito, N., On Hadamard groups, III, Kyushu J. math., 51, 1-11, (1997)
[11] Nekrasov, K.G.; Berkovich, Y.G., Finite groups with large sums of degrees of irreducible characters, Publ. math. debrecen, 33, 333-354, (1986) · Zbl 0649.20005
[12] K. G. Nekrasov, Non-nilpotent finite groups with large sums of degrees of irreducible complex characters, Rep. Kalinin Univ. 1987, 66, 70 · Zbl 0744.20014
[13] K. G. Nekrasov, Finite groups with large sums of degrees of irreducible complex characters, II, 1987, 118, 130 · Zbl 0744.20014
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