Algebraic Topology · Problem 2.1.14

Homology — Allen Hatcher

Mathematics — Topology · exact-sequences-abelian-groups

Problem

Determine whether there exists a short exact sequence

0 \to \mathbb{Z}_4 \to \mathbb{Z}_8 \oplus \mathbb{Z}_2 \to \mathbb{Z}_4 \to 0

. More generally, determine which abelian groups

A

fit into a short exact sequence

0 \to \mathbb{Z}_{p^m} \to A \to \mathbb{Z}_{p^n} \to 0

with

p

prime. What about the case of short exact sequences

0 \to \mathbb{Z} \to A \to \mathbb{Z}_n \to 0