In this paper, we prove that the concepts of cocyclic Hadamard matrix and Hadamard group are equivalent. A general procedure for constructing Hadamard groups and classifying such groups on the basis of isomorphism type is given. To illustrate the ideas, cocyclic Hadamard matrices over dihedral groups are constructed and the corresponding Hadamard groups classified.ᮊ 1997 Academic Press
Abstract. For n > 2, let Γn denote either SL(n, Z) or Sp(n, Z). We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group H ≤ Γn. This forms the main component of our methods for computing with such arithmetic groups H. More generally, we provide algorithms for computing with Zariski dense groups in Γn. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.
We present an algorithm that decides whether a finitely generated linear group over an infinite field is solvable-by-finite: a computationally effective version of the Tits alternative. We also give algorithms to decide whether the group is nilpotent-by-finite, abelian-by-finite, or central-byfinite. Our algorithms have been implemented in MAGMA and are publicly available.
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