On Hadamard groups. III.

*(English)*Zbl 0906.05013From the introduction: A group \(G(n)\) of type \(Q\) is defined by \(G(n)= \langle a,b\mid a^{4n}= e\), \(a^{2n}= b^2\) and \(b^{-1}ab= a^{-1}\rangle\). \(G(n)\) is dicyclic, has order \(8n\) and a Sylow 2-subgroup of \(G(n)\) is a generalized quaternion group. \(G(n)\) contains only one involution \(a^{2n}= b^2\), which is central in \(G(n)\).

Let \(G\) be a group of order \(8n\) containing a central involution \(e^*\). Let \(D\) be a transversal of \(G\) with respect to \(\langle e^*\rangle\). Then we have that \(G= D\cup De^*\) and \(D\cap De^*=\varnothing\). If there exists a transversal \(D\) such that \(| D\cap Da|= 2n\) for any element \(a\) of \(G\) outside \(\langle e^*\rangle\), then \(D\) and \(G\) are called an Hadamard subset and an Hadamard group respectively. If there exists an Hadamard group \(G\) of order \(8n\), then there exists an Hadamard matrix \(H\) of order \(4n\) such that the automorphism group \(\operatorname{Aut}(H)\) of \(H\) contains \(G\) as a ‘regular’ subgroup [N. Ito, J. Algebra 168, No. 3, 981-897 (1994; Zbl 0906.05012)].

Thus Hadamard groups are introduced to show the existence of Hadamard matrices which have an optimal group-theoretical property with the hope that such Hadamard matrices exist for every possible order. On the other hand, W. de Launey and K. J. Horadam [Des. Codes Cryptography 3, No. 1, 75-87 (1993; Zbl 0838.05019)] and D. L. Flannery [J. Pure Appl. Algebra 112, No. 2, 181-190 (1996; Zbl 0867.20043)] are working in a similar manner to show the existence of ‘cocyclic’ Hadamard matrices of order \(4n\) developed over groups of order \(4n\), where \(n\) is any positive integer, using the cohomology theory of groups. Furthermore, quite recently D. L. Flannery [J. Algebra 192, No. 2, 749-779 (1997; Zbl 0889.05032)] has obtained a fundamental result which tells us that the existence of an Hadamard group of order \(8n\) is equivalent to the existence of a cocyclic Hadamard matrix of order \(4n\). He also makes a detailed consideration on dihedral groups which seems to be parallel to our consideration on groups of type \(Q\).

Now let \(D\) be a transversal of \(G(n)\) with respect to \(\langle e^*\rangle\), where \(e^*= a^{2n}\). Then we list elements of \(D\) as follows: \[ ee_0, ae_1,\dots, a^{2n- 1}e_{2n- 1}; bf_0, baf_1,\dots, ba^{2n- 1}f_{2n- 1}, \] where \(e_i\) and \(f_i\) are equal to \(e\) or \(e^*\) for \(i= 0,1,\dots, 2n-1\). Now we define the polynomials \(c(x)\) and \(d(x)\) associated with \(D\) as follows: \[ c(x)= c_0+ c_1x+\cdots+ c_{2n- 1}x^{2n- 1} \] and \[ d(x)= d_0+ d_1x+\cdots+ d_{2n- 1}x^{2n- 1}, \] where \(c_i\) and \(d_j\) are equal to \(1\) or \(-1\), according to whether \(e_i\) and \(f_j\) are equal to \(e\) or \(e^*\) respectively for \(i\) and \(j=0,1,\dots, 2n-1\). Moreover, we regard \(c(x)\) and \(d(x)\) as elements of \(Z[x]/(1+ x^{2n})\). Then \(D\) is an Hadamard subset if and only if the following equality holds: \(c(x^{-1})c(x)+ d(x^{-1})d(x)= 4n\). (For this, see [N. Ito, Groups – Korea ’94. Proceedings of the international conference, Pusan, Korea, August 18-25, 1994. Berlin: Walter de Gruyter. 149-155 (1995; Zbl 0864.05021)].) We conjecture that every \(G(n)\) is an Hadamard group.

See [J. Algebra 168, No. 3, 981-987 (1994; Zbl 0906.05012)] for Part I and [J. Algebra 169, No. 3, 936-642 (1994; Zbl 0808.05016)] for Part II.

Let \(G\) be a group of order \(8n\) containing a central involution \(e^*\). Let \(D\) be a transversal of \(G\) with respect to \(\langle e^*\rangle\). Then we have that \(G= D\cup De^*\) and \(D\cap De^*=\varnothing\). If there exists a transversal \(D\) such that \(| D\cap Da|= 2n\) for any element \(a\) of \(G\) outside \(\langle e^*\rangle\), then \(D\) and \(G\) are called an Hadamard subset and an Hadamard group respectively. If there exists an Hadamard group \(G\) of order \(8n\), then there exists an Hadamard matrix \(H\) of order \(4n\) such that the automorphism group \(\operatorname{Aut}(H)\) of \(H\) contains \(G\) as a ‘regular’ subgroup [N. Ito, J. Algebra 168, No. 3, 981-897 (1994; Zbl 0906.05012)].

Thus Hadamard groups are introduced to show the existence of Hadamard matrices which have an optimal group-theoretical property with the hope that such Hadamard matrices exist for every possible order. On the other hand, W. de Launey and K. J. Horadam [Des. Codes Cryptography 3, No. 1, 75-87 (1993; Zbl 0838.05019)] and D. L. Flannery [J. Pure Appl. Algebra 112, No. 2, 181-190 (1996; Zbl 0867.20043)] are working in a similar manner to show the existence of ‘cocyclic’ Hadamard matrices of order \(4n\) developed over groups of order \(4n\), where \(n\) is any positive integer, using the cohomology theory of groups. Furthermore, quite recently D. L. Flannery [J. Algebra 192, No. 2, 749-779 (1997; Zbl 0889.05032)] has obtained a fundamental result which tells us that the existence of an Hadamard group of order \(8n\) is equivalent to the existence of a cocyclic Hadamard matrix of order \(4n\). He also makes a detailed consideration on dihedral groups which seems to be parallel to our consideration on groups of type \(Q\).

Now let \(D\) be a transversal of \(G(n)\) with respect to \(\langle e^*\rangle\), where \(e^*= a^{2n}\). Then we list elements of \(D\) as follows: \[ ee_0, ae_1,\dots, a^{2n- 1}e_{2n- 1}; bf_0, baf_1,\dots, ba^{2n- 1}f_{2n- 1}, \] where \(e_i\) and \(f_i\) are equal to \(e\) or \(e^*\) for \(i= 0,1,\dots, 2n-1\). Now we define the polynomials \(c(x)\) and \(d(x)\) associated with \(D\) as follows: \[ c(x)= c_0+ c_1x+\cdots+ c_{2n- 1}x^{2n- 1} \] and \[ d(x)= d_0+ d_1x+\cdots+ d_{2n- 1}x^{2n- 1}, \] where \(c_i\) and \(d_j\) are equal to \(1\) or \(-1\), according to whether \(e_i\) and \(f_j\) are equal to \(e\) or \(e^*\) respectively for \(i\) and \(j=0,1,\dots, 2n-1\). Moreover, we regard \(c(x)\) and \(d(x)\) as elements of \(Z[x]/(1+ x^{2n})\). Then \(D\) is an Hadamard subset if and only if the following equality holds: \(c(x^{-1})c(x)+ d(x^{-1})d(x)= 4n\). (For this, see [N. Ito, Groups – Korea ’94. Proceedings of the international conference, Pusan, Korea, August 18-25, 1994. Berlin: Walter de Gruyter. 149-155 (1995; Zbl 0864.05021)].) We conjecture that every \(G(n)\) is an Hadamard group.

See [J. Algebra 168, No. 3, 981-987 (1994; Zbl 0906.05012)] for Part I and [J. Algebra 169, No. 3, 936-642 (1994; Zbl 0808.05016)] for Part II.