Show that the multiplicative group of integers mod 35 is not isomorphic to the additive group of integers mod 24

Question

I got this question on an abstract algebra test back when I was still in undergrad. I was going over some old notes from undergrad (just for fun) and I realized I had a completely wrong answer to this question so I set about trying to solve it. I am really struggling with it. To be honest, I think I just have a hard time disproving two groups are isomorphic unless they have different cardinality.

Any suggestions?

Answer 1

Hint:

Can you show, $\forall n\in(\mathbb Z/35\mathbb Z)^\times$, $n^{12}\equiv1\bmod35$?

Answer 2

The multiplicative group $(\mathbb{Z}_{35})^{\times}$ is isomorphic to $(\mathbb{Z}_{5})^{\times} \times (\mathbb{Z}_{7})^{\times}$, which is isomorphic to the additive group $\mathbb{Z}_{4} \times \mathbb{Z}_{6}$. Those three groups are not isomorphic to $\mathbb{Z}_{24}$, because $4$ and $6$ are not relatively prime.