Finitely generated nilpotent groups with isomorphic finite quotients

Pickel, P. F.

doi:10.1090/S0002-9947-1971-0291287-3

Finitely generated nilpotent groups with isomorphic finite quotients
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by P. F. Pickel

Trans. Amer. Math. Soc. 160 (1971), 327-341

DOI: https://doi.org/10.1090/S0002-9947-1971-0291287-3

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Abstract:

Let $\mathcal {F}(G)$ denote the set of isomorphism classes of finite homomorphic images of a group $G$. We say that groups $G$ and $H$ have isomorphic finite quotients if $\mathcal {F}(G) = \mathcal {F}(H)$. In this paper we show that if $G$ is a finitely generated nilpotent group, the finitely generated nilpotent groups $H$ for which $\mathcal {F}(G) = \mathcal {F}(H)$) lie in only finitely many isomorphism classes. This is done using some finiteness results from the theory of algebraic groups along with some heretofore unpublished results of A. Borel.

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