Algebraic Topology · Problem 3F.4

Cohomology — Allen Hatcher

Mathematics — Topology · divisible-groups-ext-vanishing

Problem

An abelian group GG is defined to be divisible if the map G→nGG \xrightarrow{n} G, g↦ngg \mapsto ng, is surjective for all n>1n > 1. Show that a group is divisible iff it is a quotient of a direct sum of Q\mathbb{Q}'s. Deduce from the previous problem that if GG is divisible then Ext(A,G)=0\mathrm{Ext}(A, G) = 0 for all AA.

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