Number of elements in a set of nth roots of unity I am confused about number of nth roots of unity because I know that set of nth roots of unity is a finite abelian group but somewhere I read for some n ,this is infinite group. Please tell me about these cases and whether n belongs to $\Bbb N$ or $\Bbb Z$
 A: It is not quite clear from your phrasing which of these two cases you are talking about.
Case 1. For a particular $n \in \mathbb{N}$, consider the set of $n^\text{th}$ roots of unity under the operation of multiplication. This forms an abelian group with $n$ elements. For example, if $n = 4$, then you have the finite (four element) abelian group $\{1, -1, i, -i\}$ under multiplication.
Case 2. Consider the set of all roots of unity under the operation of multiplication. Again, this forms a group, but it has infinitely many elements. (It includes the first root of unity, the second roots of unity, the third roots of unity, etc.; essentially, it is the union of all such groups as described in the first case.) 
A: The n-th roots of unity are best understood using the polar form of complex numbers and the exponential function. The n-th roots are $e^{2\pi ik/n}$ with $k\in\mathbb{N}$ and $0\leq k<n$. Geometrically these are the points of a regular n-gon on the unit circle in the complex plane with $1$ as a fixed point. From the polar form we can see that multiplying two of them together will add the $k_1$ and $k_2$ components together, and reducing them modulo $n$ will give us the polar form of another n-th root of unity. Because the multiplication of complex numbers is commutative, $1$ is a member of the n-th roots and $e^{2\pi i/n}$ generates the group it is an cyclic group of order $n$. There is no $n$ for which this is an infinite group.
However the entire unit circle in the complex plane forms a group and the n-th roots are a subgroup. I can also construct a group isomorphic to $\mathbb{Z}$ by taking an irrational number $q$ and generating the group with $e^{2\pi iq}$ and it's inverse. Maybe these are the infinite cases you've heard about.
