Abstract
In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups G are fully characterized in the class of all groups by the set tp(G) of types realized in G. Furthermore, it turns out that these groups G are fully characterized already by some particular rather restricted fragments of the types from tp(G). In particular, every finitely generated nilpotent group is completely defined by its ∃+-types, while a finitely generated rigid group is completely defined by its ∀-types, and a finitely generated metabelian or polycyclic group is completely defined by its ∀∃-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.
| Language | English (US) |
|---|---|
| Pages | 1613-1632 |
| Number of pages | 20 |
| Journal | International Journal of Algebra and Computation |
| Volume | 28 |
| Issue number | 8 |
| DOIs | |
| State | Published - Dec 1 2018 |
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Characterization of finitely generated groups by types. / Miasnikov, Alexei; Romanovskii, N. S.
In: International Journal of Algebra and Computation, Vol. 28, No. 8, 01.12.2018, p. 1613-1632.Research output: Contribution to journal › Article
TY - JOUR
T1 - Characterization of finitely generated groups by types
AU - Miasnikov, Alexei
AU - Romanovskii, N. S.
PY - 2018/12/1
Y1 - 2018/12/1
N2 - In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups G are fully characterized in the class of all groups by the set tp(G) of types realized in G. Furthermore, it turns out that these groups G are fully characterized already by some particular rather restricted fragments of the types from tp(G). In particular, every finitely generated nilpotent group is completely defined by its ∃+-types, while a finitely generated rigid group is completely defined by its ∀-types, and a finitely generated metabelian or polycyclic group is completely defined by its ∀∃-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.
AB - In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups G are fully characterized in the class of all groups by the set tp(G) of types realized in G. Furthermore, it turns out that these groups G are fully characterized already by some particular rather restricted fragments of the types from tp(G). In particular, every finitely generated nilpotent group is completely defined by its ∃+-types, while a finitely generated rigid group is completely defined by its ∀-types, and a finitely generated metabelian or polycyclic group is completely defined by its ∀∃-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.
UR - http://www.scopus.com/inward/record.url?scp=85054823530&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85054823530&partnerID=8YFLogxK
U2 - https://doi.org/10.1142/S0218196718400118
DO - https://doi.org/10.1142/S0218196718400118
M3 - Article
VL - 28
SP - 1613
EP - 1632
JO - International Journal of Algebra and Computation
T2 - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
SN - 0218-1967
IS - 8
ER -