Rationality of verbal subsets in solvable groups. (English. Russian original) Zbl 1485.20086
Algebra Logic 57, No. 1, 39-48 (2018); translation from Algebra Logika 57, No. 1, 57-72 (2018).
Summary: A verbal subset of a group \(G\) is a set \(w[G]\) of all values of a group word \(w\) in this group. We consider the question whether verbal subsets of solvable groups are rational in the sense of formal language theory. It is proved that every verbal subset \(w[N]\) of a finitely generated nilpotent group \(N\) with respect to a word \(w\) with positive exponent is rational. Also we point out examples of verbal subsets of finitely generated metabelian groups that are not rational.
MSC:
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
| 20F16 | Solvable groups, supersolvable groups |
| 20F18 | Nilpotent groups |
| 20F05 | Generators, relations, and presentations of groups |
References:
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