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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
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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

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