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Isomorphism classes and derived series of certain almost-free groups. (English) Zbl 0837.20044

For a group \(G\) denote by \(\gamma_n G\) (correspondingly, by \(G^{(n)}\)) the \(n\)th member of its lower central series (derived series). Recall that \(\gamma_1 G=G=G^{(1)}\), \(\gamma_{n+1} G=[\gamma_n G, G]\) and \(G^{(n+1)}=[G^{(n)}, G^{(n)}]\) for \(n\geq 1\). G. Baumslag [Bull. Am. Math. Soc. 73, 621-622 (1967; Zbl 0153.35001)] introduced a class of groups \(G_{ij}=\langle a, b, c\mid a=[c^i,a]\cdot [c^j,b]\rangle\); \(i,j\in \mathbb{Z}\). He has proved that those groups, though not being free, closely resemble the free group \(F_2\) on two generators. Namely, there hold \(G_{ij}/ \gamma_n G_{ij} \cong F_2/\gamma_n F_2\) (for all \(n \in \mathbb{N})\), and \(\bigcap_{n \geq 1} \gamma_n G_{ij}=(1)=\bigcap_{n \geq 1} \gamma_n F_2\). It has been proved also that these groups \(G_{ij}\) have the same (for all \(i, j \in \mathbb{Z}\)) quotients \(G_{ij}/\gamma_n G_{ij}\) and \(G_{ij}/G^{(k)}_{ij}\) for \(k=2, 3\).
In this paper it is shown that there exist several distinct isomorphism types among the groups \(G_{ij}\). Yet, it remains unknown whether all \(G_{ij}\) are distinct or not. Also, it has been ‘computed’ by the authors that for some pairs \((i,j)\) the quotients \(G_{ij}/G^{(k)}_{ij}\) for \(k \geq 4\) can differ from the corresponding quotients for \(F_2\). To obtain these results the authors have used computer-aided determination of the number of homomorphisms in the set \(\operatorname{Hom}(G_{ij}, \text{GL}(2, \mathbb{Z}_m))\) for several pairs \((i,j)\) and \(m \in \mathbb{N}\).

MSC:

20F14 Derived series, central series, and generalizations for groups
20F05 Generators, relations, and presentations of groups
20E34 General structure theorems for groups
20E05 Free nonabelian groups

Citations:

Zbl 0153.35001
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References:

[1] Baumslag G., Bull. Amer. Math. Soc. 73 pp 621– (1967) · Zbl 0153.35001
[2] Havas G., Computational Group Theory: Durham, 1982 (1984)
[3] Holt D., Groups, Combinatorics and Geometry: Durham, 1990 (1990)
[4] Magnus W., Combinatorial Group Theory, (1976)
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