Direct factors of profinite completions and decidability

Details Direct factors of profinite completions and decidability Direct factors of profinite completions and decidability

2008-10-02

arxiv.org/abs/0810.0399v1

Mathematics

Group Theory

To appear in the Journal of Group Theory

Scientific paper

We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$.

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