Direct products and profinite completions - Nikolov, N; Segal, D (2007)
Let G be a finitely generated residually finite group and A a finitely generated normal subgroup. Then G and A are naturally embedded in their respective profinite completions and . The inclusion A G induces a morphism (continuous homomorphism) : , and the image of is the closure of A in . If A happens to be a direct factor of G, i.e. G A B for some normal subgroup B, then the profinite topology on G induces the profinite topology on A, so that the morphism is injective and may be used to identify with ; and . In [D. Goldstein and R. M. Guralnick. The direct product theorem for profinite groups. J. Group Theory 9 (2006), 317322, Question 3.1], Goldstein and Guralnick ask whether the converse holds: if is a direct factor of , does it follow that A is a direct factor of G? (We take the hypothesis to mean: is injective and is a direct factor of .) We show that the answer is no in general, but yes in a suitably restricted category, namely the virtually polycyclic groups. © Walter de Gruyter 2007.