I have these three groups and I was wondering if they were isomorphic to each other? $$a)\space \mathbb{Z}_{60}\space b)\space \mathbb{Z}_{6} \times \mathbb{Z}_{10} \space c)\space\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{15}$$
So first I know that c) is isomorphic to a) because gcd(2,2,15)=1 so it is cyclic and the order of both of them is $60$, however I am not sure how to go further with the relationship with the rest of them? Can anyone show me how! Cheers
Actually, $\mathbb Z_6\times\mathbb Z_{10}$ and $\mathbb Z_2\times\mathbb Z_2\times\mathbb Z_{15}$ are isomorphic (and none of them is cyclic), since both of them are isomorphic to $\mathbb Z_2\times\mathbb Z_2\times\mathbb Z_3\times\mathbb Z_5$.
On the other hand, $\mathbb Z_{60}$ has an element whose order is $60$ (that is, it is a cyclic group), whereas none of the other two has such an element. Therefore, $\mathbb Z_{60}$ is not isomorphic to any of the other two.