Can a ℤ-module always be extended to a ℚ-module?

[novel matrix]

Basically the title. I'm trying to either show that any finite abelian group 𝐺 considered as a ℤ-module can be extended to a ℚ-module or find a counterexample.

It seems like just letting 𝑀 be formal ℚ-linear finite sums of elements of 𝐺 (and identifying things like 𝑎𝑔₁ + 𝑎𝑔₂ with 𝑎𝑔₁𝑔₂ where 𝑎∈ℚ) gives a module, but I'm afraid it doesn't extend 𝐺 because it collapses something it shouldn't or somewhere in there.

[worldly sorrel]

1. Do not ping the Moderators, unless someone is breaking the rules.
2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
3. Do not ping other members randomly for help.
4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
5. Wait patiently for a helper to come along.
6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[novel matrix]

It seems like we could potentially let 𝑀 be the regular ℚ𝐺-module then quotient 𝑀 by things we want to be the same? I'm not sure; this exercise is confusing

+close

[hexed oar]

Do you still want to close your help request?

No eligible helpers were found in this thread. You can still close this post if you don't require helper any longer.