# Hatcher 2.1.14

Hatcher, Algebraic Topology, Chapter 2, Section 1

14. 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 the 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 $0\to\mathbb{Z}\to A\to\mathbb{Z}_n\to 0$?
Proof.

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