Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T01:25:43.974Z Has data issue: false hasContentIssue false

Symmetric powers, metabelian Lie powers and torsion in groups

Published online by Cambridge University Press:  24 October 2008

Ralph Stöhr
Affiliation:
Department of Mathematics, UMIST, P.O. Box 88, Manchester, M60 1QD

Extract

In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free group

where Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Hannebauer, T. and Stöhr, R.. Homology of groups with coefficients in free metabelian Lie powers and exterior powers of relation modules and applications to group theory. In: Proceedings of the Second International Group Theory Conference, Bressanone 1989, Rend. Circ.Mat. Palermo (2) Suppl. 23 (1990), 77113.Google Scholar
[2]Hartley, B. and Stöhr, R.. A note on the homology of free abelianized extensions. Proc. Amer. Math. Soc. 113 (1991), 923932.CrossRefGoogle Scholar
[3]Hartley, B. and Stöhr, R.. Homology of higher relation modules and torsion in free central extensions of groups. Proc. London Math. Soc. 62 (1991), 325352.CrossRefGoogle Scholar
[4]Hilton, P. and Stammbach, U.. A Course in Homological Algebra (Springer, 1971).CrossRefGoogle Scholar
[5]Kovacs, L. G., Kuz'min, Yu. V. and Stöhr, R.. Homology of free abelianized extensions of groups. Mat. Sb. 182 (1991), 526542 (in Russian, English translation: Math. USSR Sbornik 72 (1992), 503518).Google Scholar
[6]Shmel'kin, A. L.. Wreath products and varieties of groups. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 149170 (in Russian).Google Scholar
[7]Shmel'kin, A. L. and Stöhr, R.. On torsion in certain free centre-by-soluble groups. J. Pure Appl. Algebra 88 (1993), 225237.CrossRefGoogle Scholar
[8]Stöhr, R.. Homology of metabelian Lie powers and torsion in relatively free groups. Quart. J. Math. (Oxford) 43 (1992), 361380.CrossRefGoogle Scholar
[9]Stöhr, R.. Homology of free Lie powers and torsion in groups. Israel J. Math. 84 (1993), 6587.CrossRefGoogle Scholar