Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-16T23:30:40.296Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Theorem of Bovdi

Published online by Cambridge University Press:  20 November 2018

M, M. Parmenter
M, M. Parmenter*
Affiliation:
University of Alberta, Edmonton, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If p is a prime, we call an element x ≠ 1 of a group G a generalized p-element if, for every n ≧ 1, there exists r ≧ 0 such that xpr ∈ Gn, where Gn is the nth term of the lower central series of G. Bovdi [1] proved that if G is a finitely generated group having a generalized p-element, and if ∩nΔn(Z(G) = 0 where Δ(Z(G)) is the augmentation ideal, then G is residually a finite p-group.

We recall that if R is a ring, then the nth dimension subgroup of G over R, denoted by Dn(R(G)), is defined to be {g | g – 1 ∈ Δn(R(G))}. In this note, we show that if G is finitely generated, then ∩nDn(Zp∧(G)) = 1 ⇔ ∩nΔn(Zp∧ (G)) = 0 ⇔ G is residually a finite p-group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 929 - 932
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

This work will form part of the author's Ph.D. thesis, written at the University of Alberta under the direction of Professor S. K. Sehgal.

References

1. Bovdi, A. A., The intersection of the powers of the fundamental ideal of an integral group ring, Mat. Zametki 2 (1967), 129–132.Google Scholar
2. Hall, P., Nilpotent groups (Canad. Math. Congress, University of Alberta, 1957).Google Scholar
3. Hartley, B., The residual nilpotence of wreath products, Proc. London Math. Soc. 20 (1970), 365–392.Google Scholar
4. Mital, J. N., On residual nilpotence, J. London Math. Soc. 2 (1970), 337–345.Google Scholar
5. Passi, I. B. S., Polynomial maps on groups, J. Algebra 9 (1968), 121–151.Google Scholar
6. Passi, I. B. S., Dimension subgroups, J. Algebra 9 (1968), 152–182.Google Scholar
7. Sandling, R., The modular group rings of p-groups, Ph.D. Thesis, University of Chicago, 1969.Google Scholar