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Computing polycyclic presentations for polycyclic rational matrix groups. (English) Zbl 1124.20306

Summary: We describe practical algorithms for computing a polycyclic presentation and for facilitating a membership test for a polycyclic subgroup of \(\text{GL}(d,\mathbb{Q})\). A variation of this method can be used to check whether a finitely generated subgroup of \(\text{GL}(d,\mathbb{Q})\) is solvable or solvable-by-finite. We report on our implementations of the algorithms for determining a polycyclic presentation and checking solvability.

MSC:

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
20-04 Software, source code, etc. for problems pertaining to group theory
68W30 Symbolic computation and algebraic computation
20E07 Subgroup theorems; subgroup growth
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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References:

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