416 research outputs found
Connected quandles and transitive groups
We establish a canonical correspondence between connected quandles and
certain configurations in transitive groups, called quandle envelopes. This
correspondence allows us to efficiently enumerate connected quandles of small
orders, and present new proofs concerning connected quandles of order p and 2p.
We also present a new characterization of connected quandles that are affine
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Algebra, matrices, and computers
What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix groups and new methods for handling matrix groups in a computer
Algebra, matrices, and computers
What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix groups and new methods for handling matrix groups in a computer
Zariski density and computing with -integral groups
We generalize our methodology for computing with Zariski dense subgroups of
and , to accommodate
input dense subgroups of and . A key task, backgrounded by the Strong Approximation theorem, is
computing a minimal congruence overgroup of . Once we have this overgroup,
we may describe all congruence quotients of . The case receives
particular attention
Zariski density and computing in arithmetic groups
For n > 2, let Gamma _n 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\leq \Gamma _n. 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 Gamma _n. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups
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