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An \(L_2\)-quotient algorithm for finitely presented groups on arbitrarily many generators. (English) Zbl 1315.20034
Summary: We generalize the Plesken-Fabiańska \(L_2\)-quotient algorithm [W. Plesken and A. Fabiańska, J. Algebra 322, No. 3, 914-935 (2009; Zbl 1253.20033)] for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the invariant ring of \(\mathrm{GL}(2,K)\) on \(m\) copies of \(\mathrm{SL}(2,K)\) by simultaneous conjugation. By giving this description, we generalize and simplify some of the known results in invariant theory. An implementation of the algorithm is available in the computer algebra system Magma.

MSC:
20F05 Generators, relations, and presentations of groups
13A50 Actions of groups on commutative rings; invariant theory
20-04 Software, source code, etc. for problems pertaining to group theory
68W30 Symbolic computation and algebraic computation
Software:
QuillenSuslin; Magma
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