On Hadamard groups.

*(English)*Zbl 0906.05012From 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 \(G\).

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 |