Groups whose proper quotients are finite-by-nilpotent. (English) Zbl 0725.20027
A group \(G\) is said to be an \(FN_ c\)-group if the \((c+1)\)-st term \(\gamma_{c+1}G\) of its lower central series is finite. A group \(G\) is called a \(JNFN_ c\)-group if all its proper quotients are FN\(_ c\)-groups, but \(G\) itself is not. The structure of JNFA-groups \((JNFN_ c\)-groups with \(c=1)\) has been described by D. J. S. Robinson and the author [J. Algebra 118, No. 2, 346-368 (1988; Zbl 0658.20019)]. In this paper we study JNFN\(_ c\)-groups with non-trivial Fitting subgroup for arbitrary \(c\) which are naturally divided into two classes.
It turns out that \(JNFN_ c\)-groups with non-trivial centre are just the torsion-free nilpotent groups \(G\) of class \(c+1\) such that \(\gamma_{c+1}G\) is infinite cyclic and \(Z(G)\) is a subgroup of the rationals. Next, we deal with \(JNFN_ c\)-groups with trivial centre and non-trivial Fitting subgroup \(A\). It is shown that \(A\) is abelian and then \(A\) is a faithful just infinite module for the non-trivial \(FN_ c\)-group \(Q=:G/A\).
The investigation also shows that it is significantly harder to describe \(JNFN_ c\)-groups with \(c\geq 2\) than is the case \(c=1\); some interesting but complicated problems about groups, modules and cohomology and many new techniques are involved.
In the final section we consider faithful just infinite modules over nilpotent \(FN_ c\)-groups. For this purpose a well-known theorem of P. Hall [Proc. Camb. Philos. Soc. 52, 611-616 (1956; Zbl 0072.25801)] is extended in the following form: If \(G\) is an \(FN_ c\)-group, then \(G/C_ G(\gamma_ cG)\) is residually finite and of finite exponent. Moreover, \(G/Z_{2c-2}(G)\) is of finite exponent, provided \(c\geq 2\). By using this proposition, necessary and sufficient conditions are obtained for a nontrivial nilpotent \(FN_ c\)-group \(Q\) to have a faithful just infinite module, where the crux of the matter is the following theorem on modules: Let \(Q\) be a nilpotent group with centre \(K\). If \(Q\) has infinite torsion-free rank and the torsion subgroup of \(K\) is a locally cyclic \(p'\)-group (\(p\geq 0\)), then \(Q\) has a faithful just infinite module of characteristic \(p\) that is simple. These results are markedly different from those in the case \(c=1\).
It turns out that \(JNFN_ c\)-groups with non-trivial centre are just the torsion-free nilpotent groups \(G\) of class \(c+1\) such that \(\gamma_{c+1}G\) is infinite cyclic and \(Z(G)\) is a subgroup of the rationals. Next, we deal with \(JNFN_ c\)-groups with trivial centre and non-trivial Fitting subgroup \(A\). It is shown that \(A\) is abelian and then \(A\) is a faithful just infinite module for the non-trivial \(FN_ c\)-group \(Q=:G/A\).
The investigation also shows that it is significantly harder to describe \(JNFN_ c\)-groups with \(c\geq 2\) than is the case \(c=1\); some interesting but complicated problems about groups, modules and cohomology and many new techniques are involved.
In the final section we consider faithful just infinite modules over nilpotent \(FN_ c\)-groups. For this purpose a well-known theorem of P. Hall [Proc. Camb. Philos. Soc. 52, 611-616 (1956; Zbl 0072.25801)] is extended in the following form: If \(G\) is an \(FN_ c\)-group, then \(G/C_ G(\gamma_ cG)\) is residually finite and of finite exponent. Moreover, \(G/Z_{2c-2}(G)\) is of finite exponent, provided \(c\geq 2\). By using this proposition, necessary and sufficient conditions are obtained for a nontrivial nilpotent \(FN_ c\)-group \(Q\) to have a faithful just infinite module, where the crux of the matter is the following theorem on modules: Let \(Q\) be a nilpotent group with centre \(K\). If \(Q\) has infinite torsion-free rank and the torsion subgroup of \(K\) is a locally cyclic \(p'\)-group (\(p\geq 0\)), then \(Q\) has a faithful just infinite module of characteristic \(p\) that is simple. These results are markedly different from those in the case \(c=1\).
Reviewer: Zhang Zhirang (Chengdu)
MSC:
| 20F18 | Nilpotent groups |
| 20F14 | Derived series, central series, and generalizations for groups |
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
| 20C12 | Integral representations of infinite groups |
| 20E34 | General structure theorems for groups |
Keywords:
lower central series; \(FN_c\)-groups; JNFA-groups; \(JNFN_c\)-groups; Fitting subgroup; torsion-free nilpotent groups; faithful just infinite modules; residually finite; finite exponent; torsion-free rank; torsion subgroup; locally cyclic \(p'\)-groupsReferences:
| [1] | K. A. Brown, Primitive group rings of soluble groups. Arch. Math.36, 404-413 (1981). · Zbl 0464.16009 · doi:10.1007/BF01223718 |
| [2] | P. Hall, Finite-by-nilpotent groups. Proc. Cambridge Philos. Soc.52, 611-616 (1956). · Zbl 0072.25801 · doi:10.1017/S0305004100031662 |
| [3] | D. J. S. Robinson, The vanishing of certain homology and cohomology groups. J. Pure Appl. Algebra7, 145-167 (1976). · Zbl 0329.20032 · doi:10.1016/0022-4049(76)90029-3 |
| [4] | D. J. S. Robinson andZ. R. Zhang, Groups whose proper quotients have finite derived sub-groups. J. Algebra (2)118, 346-368 (1988). · Zbl 0658.20019 · doi:10.1016/0021-8693(88)90026-9 |
| [5] | D. J. S.Robinson, A Course in the Theory of Groups. Berlin-Heidelberg-New York 1982. · Zbl 0483.20001 |
| [6] | D. J. S. Robinson andJ. S. Wilson, Soluble groups with many polycyclic quotients. Proc. London Math. Soc. (3)48, 193-229 (1984). · doi:10.1112/plms/s3-48.2.193 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
