On a Theorem of Bovdi

Parmenter, M, M.

Canadian journal of mathematics, Tome 23 (1971) no. 5, pp. 929-932

Résumé

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 ∩n Δn (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 ∩n Dn (Z p ∧(G)) = 1 ⇔ ∩n Δn (Zp ∧ (G)) = 0 ⇔ G is residually a finite p-group.

Détail

DOI : 10.4153/CJM-1971-101-x

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