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The Andrews-Curtis problem for \(\mathcal F (\mathfrak M)\). (English. Russian original) Zbl 0743.20026

Math. Notes 48, No. 1, 671-672 (1990); translation from Mat. Zametki 48, No. 1, 75-77 (1990).
The following theorem is proved. Let \(F\) be a free group with basis \(\{x_ 1,\dots,x_ n\}\) \((n\geq 2)\), let \(H\) be a normal subgroup of \(F\) contained in the commutator subgroup \(F'\) and let \(U\) be a normal subgroup of \(F\) contained in \(H\) and containing the verbal subgroup \(V(H)\). If the group \(T\) of extended Nielsen transformations acts transitively on \(F/H\), then \(T\) acts transitively also on \(F/U\). Some consequences for the solution of the Andrews-Curtis problem in a variety are given.

MSC:

20E10 Quasivarieties and varieties of groups
20E05 Free nonabelian groups
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References:

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