Hopficity of profinite completions of abelian groups

2026/07/21 by M. Brescia, E. Ingrosso, M. Trombetti
#math.GR

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Abstract

We determine exactly when the profinite completion of an arbitrary abelian group is topologically Hopfian. For an abelian group A, we prove that \widehat A is topologically Hopfian \Longleftrightarrow A/pA is finite for every prime p. As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes q_i, the group \bigoplus_i≥1\Z[1/q_i] is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.

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