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On Hadamard groups. III. (English) Zbl 0906.05013
From 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.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups