- #1
Mr Davis 97
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Homework Statement
Are the groups ##\mathbb{Z}_8 \times \mathbb{Z}_{10} \times \mathbb{Z}_{24}## and ##\mathbb{Z}_4 \times \mathbb{Z}_{12} \times \mathbb{Z}_{40}## isomorphic? Why or why not?
Homework Equations
The Attempt at a Solution
I think I am misunderstanding the Theorem of Finitely Generated Abelian Groups, because to me it seems that we can just decompose each direct product into ##\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5##, and so they are isomorphic. Why can't this be done, and why are they not isomorphic?