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On Hadamard groups. (English) Zbl 0906.05012
From the introduction: An Hadamard matrix $$H$$ of order $$n$$ is a $$(-1,1)$$-matrix of degree $$n$$ such that $$HH^t= nI$$, where $$t$$ denotes the transposition and $$I$$ is the identity matrix of degree $$n$$. An automorphism of $$H$$ is a pair of signed permutations $$R$$ and $$S$$ such that $$H= RHS$$, and the set of all automorphisms of $$H$$ forms a group which is called the automorphism group of $$H$$ and is denoted by $$\operatorname{Aut}(H)$$. In order to investigate the structure of $$\operatorname{Aut}(H)$$ it is convenient to regard $$H$$ as a kind of incidence matrix of a block design which is defined as follows.
Let $$D= (P, B)$$ be a block design, where $$P$$ and $$B$$ are the sets of points and blocks, respectively, satisfying the following conditions: (1) $$| P|=| B|= 2n$$, where $$| X|$$ denotes the number of elements in a finite set $$X$$. For $$\underline a\in B$$ we have that $$|\underline a|= n$$ and $$P-\underline a\in B$$. (2) For $$\underline a,\underline b\in B$$ we have that $$|\underline a\cap\underline b|= n/2$$, provided that $$\underline b\neq\underline a$$ and $$P-\underline a$$. (3) We may put $$P= \{a_1,\dots, a_n,b_1,\dots, b_n\}$$ so that $$|\underline a\cap\{a_i, b_i\}|= 1$$ for any $$\underline a\in B$$ and $$1\leq i\leq n$$. We call such a $$D$$ an Hadamard design of order $$2n$$.
Label blocks of $$B$$ so that $$B= \{\underline a_1,\dots,\underline a_n, P-\underline a_1,\dots, P-\underline a_n\}$$. Then the relation between $$D$$ and $$H$$ is as follows. $$\underline a_i$$ contains $$a_j$$ or $$b_j$$ according to whether $$H(i,j)= 1$$ or $$-1$$, where $$H(i, j)$$ is the $$(i,j)$$ component of $$H$$. It is easily seen that $$\operatorname{Aut}(H)$$ can be identified with the automorphism group $$\operatorname{Aut}(D)$$ of $$D$$ in a natural way.
From now on we consider the case where $$\operatorname{Aut}(D)$$ contains a regular subgroup which we formulate as an abstract group as follows. Let $$G$$ be a finite group of order $$2n$$. Then $$G$$ is called an Hadamard group if $$G$$ contains a subset $$D$$ with $$| D|= n$$ and an element $$e^*$$ such that (4) $$| D\cap Da|= n$$ if $$a= e$$ where $$e$$ denotes the identity element of $$G$$, $$=0$$ if $$a= e^*$$, and $$= n/2$$ for any other element $$a$$ of $$G$$, and (5) $$| Da\cap\{b, be^*\}|= 1$$ for any elements $$a$$ and $$b$$ of $$G$$.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B05 Combinatorial aspects of block designs 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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