On Hadamard property of a certain class of finite groups.

*(English)*Zbl 0911.20008A 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).

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).

Reviewer: Yakolev Berkovich (Afula)

##### 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.) |

##### Keywords:

finite groups; central involutions; transversals; irreducible characters; generalized quaternion groups; Hadamard groups; Hadamard 2-groups
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\textit{J. R. Cho} et al., J. Algebra 204, No. 2, 666--674 (1998; Zbl 0911.20008)

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##### References:

[1] | Baliga, A.; Horadam, K.J., Cocyclic Hadamard matrices overZ_t×Z22, 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 |

[8] | Ito, N., Remarks on Hadamard groups, Kyushu J. math., 50, 83-91, (1996) · Zbl 0889.05033 |

[9] | Ito, N., Remarks on Hadamard groups, II, Meijo U. sci. rep. fac. sci. eng., 37, 1-7, (1997) · Zbl 0892.20016 |

[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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