A residual property of certain linear groups

Stebe, Peter F.

doi:10.1090/S0002-9947-1987-0887513-1

A residual property of certain linear groups
HTML articles powered by AMS MathViewer

by Peter F. Stebe PDF

Trans. Amer. Math. Soc. 302 (1987), 333-340 Request permission

Abstract:

An extension of residual finiteness, residual finiteness with respect to nests, is demonstrated for certain subgroups of $GL(n,Z)$, the polycyclic by finite groups. It is also shown that groups containing a free subgroup of rank greater than $1$ cannot have the property. It is not settled whether or not there are other solvable by finite groups, subgroups allowed by Tits’ theorem, that are residually finite with respect to nests.

References

Gilbert Baumslag, Lecture notes on nilpotent groups, Regional Conference Series in Mathematics, No. 2, American Mathematical Society, Providence, R.I., 1971. MR 0283082
Edward Formanek, Conjugate separability in polycyclic groups, J. Algebra 42 (1976), no. 1, 1–10. MR 419608, DOI 10.1016/0021-8693(76)90021-1
Marshall Hall Jr., A topology for free groups and related groups, Ann. of Math. (2) 52 (1950), 127–139. MR 36767, DOI 10.2307/1969513
P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419–436. MR 72873, DOI 10.1112/plms/s3-4.1.419

On infinite soluble groups

44

On infinite soluble groups

44

K. A. Hirsch, On infinite soluble groups. III, Proc. London Math. Soc. (2) 49 (1946), 184–194. MR 17281, DOI 10.1112/plms/s2-49.3.184
A. I. Mal′cev, On some classes of infinite soluble groups, Mat. Sbornik N.S. 28(70) (1951), 567–588 (Russian). MR 0043088
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
A. Włodzimierz Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59 (1966), 123–135. MR 224693, DOI 10.4064/fm-59-2-123-135
B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. MR 62741, DOI 10.1098/rsta.1954.0007
V. N. Remeslennikov, Conjugacy in polycyclic groups, Algebra i Logika 8 (1969), 712–725 (Russian). MR 0280593
P. F. Stebe, A residual property of certain groups, Proc. Amer. Math. Soc. 26 (1970), 37–42. MR 260874, DOI 10.1090/S0002-9939-1970-0260874-5
Peter F. Stebe, Residual solvability of an equation in nilpotent groups, Proc. Amer. Math. Soc. 54 (1976), 57–58. MR 387410, DOI 10.1090/S0002-9939-1976-0387410-2
Peter F. Stebe, Nests in nilpotent groups, Houston J. Math. 2 (1976), no. 3, 419–426. MR 412277
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0