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Let $A,B,C$ be finitely generated abelian groups, and $A\oplus B\cong A\oplus C$. Prove that $B\cong C$.

I know that it follows from the fundamental theorem of finite abelian groups, but I have difficulty with formalizing a proof

daPollak
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1 Answers1

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Hints:

Fix decompositions for $A,B,C$ and note that they "stack" to become decompositions of $A\oplus B$ and $A\oplus C$. The components of the decompositions must line up exactly, and without loss of generality, we can line up the components of $A$ on both sides because they're identical. What does this say about the remaining components?

rschwieb
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