Let G be a group with $\displaystyle |G|=30$ .Prove that G is izomorf with the group of 30-th roots of unity .
If a group has order 30 there are 4 unique groups (up to isomorphism) to which it could be isomorphic.
Abelian
1) $\displaystyle \mathbb{Z}_{30}$ this is the one you are looking for
Nonabelian
2) $\displaystyle D_{15}$ (Dihedral group)
3) $\displaystyle D_5 \times Z_3$
4) $\displaystyle D_3 \times Z_5$
If its abelian its easy to see why it must be cyclic by the fundamental theorem of finitely generated abelian groups, and noting $\displaystyle 30=2*3*5$
hope this helps