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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 \(G\).

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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