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Note on Hadamard matrices of type Q. (English) Zbl 0537.05012
Let u be an odd integer, $$n=4u$$ and $$H=(h_{ij})$$ a Hadamard matrix of order n. Let P be the set of 2n points $$1,2,...,n,1^*,2^*,...,n^*$$. Define an n-subset $$\alpha_ i$$ of P by $$j\in a_ i$$ if $$h_{ij}=+1$$, $$j^*\in a_ i$$ if $$h_{ij}=-1$$, 1$$\leq i$$, $$j\leq n$$. $$\alpha^*_ i=P-\alpha_ i$$. $$\alpha_ i$$ and $$\alpha^*_ i$$ are blocks and B is the set of 2n blocks $$\alpha_ 1,\alpha_ 2,...,\alpha^*_ 1,\alpha^*_ 2,...,\alpha^*_ n$$. $$M(H)=(P,B)$$ is the matrix design of H. Let G be the set of all permutations $$\sigma$$ on P which induce a permutation on B, then G is the automorphism group M(H) and isomorphic to the automorphism group of H. The paper defines R to be the permutation group on P generated by three given permutations. R has order 8u, a cyclic normal subgroup of order u, a Sylow 2-subgroup which is isomorphic to the quaternion group and is regular transitive on P. H is called a Hadamard matrix of type Q if R is regular transitive on B. The paper proves the following pretty results: 1) The Hadamard matrix of quadratic residue type of order $$n=4u=q+1$$, $$q\equiv 3(mod 8)$$, q a prime power is a Hadamard matrix of type Q; 2) a Hadamard matrix of Williamson type using symmetric circulants is a Hadamard matrix of type Q. In particular, its automorphism group contains a regular transitive subgroup; 3) a Hadamard matrix of Paley type of order $$n=4u=2(q+1)$$, $$q\equiv 1(mod 4)$$, q a prime power is a Hadamard matrix of type Q.
Reviewer: Jennifer Seberry

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures