Comments on Strongly Torsion-free Groups

In the present note, we discuss certain observations made by the author in February 2009 concerning strongly torsion-free profinite groups [cf. [Mzk2], Definition 1.1, (iii)]. These observations grew out of e-mail correspondences between the author, Akio Tamagawa, and Marco Boggi, as well as oral discussions between the author and Akio Tamagawa. Definition 1. Let G be a profinite group. (i) We shall say that G is ab-torsion-free if, for every open subgroup H ⊆ G, the abelianization H ab of H is torsion-free [cf. Remark 1.1 below]. (ii) We shall say that G is ab-faithful if, for every open subgroup H ⊆ G, and every normal open subgroup N ⊆ H of H, the natural homomorphism H/N → Aut(N ab) arising from conjugation is injective. Remark 1.1. Note that G is strongly torsion-free in the sense of [Mzk2], Definition 1.1, (iii), if and only if it is topologically finitely generated and ab-torsion-free in the sense of Definition 1, (i). Remark 1.2. One verifies immediately that if G is ab-faithful, then it is slim. Indeed, this is precisely the approach taken to verifying slimness in the proof of [Mzk2], Proposition 1.4. Remark 1.3. It follows from Examples 3, 5 below that neither of the implications " ab-torsion-free =⇒ ab-faithful " , " ab-faithful =⇒ ab-torsion-free " holds. Remark 1.4. It is immediate from the definitions that every open subgroup of an ab-torsion-free (respectively, ab-faithful) profinite group is itself ab-torsion-free (respectively, ab-faithful). Proposition 2. (Automorphisms Induced on Abelianizations) Let G be a topologically finitely generated profinite group that satisfies at least one of the following two conditions: