Let n belongsto N and let zetan z belongsto C zn 1 Show z

Let n belongsto N and let zeta(n) = {z belongsto C | z^n = 1}. Show zeta(n) is a group under complex multiplication. Show zeta(n) = {e^2k pi i/n | k = 0, 1,...,(n - 1)}. Show there is a bijection phi: Z_n rightarrow zeta(n) so that phi([a] + [b]) = phi([a])phi([b]).

Solution

Closer property:

Any two number belongs to given set,it’s multiplication is also belongs to given set.

Associative property:

a,b,c   belongs to given set ,

(a*b)*c=a*(b*c).

Identity property:

If a is any number belongs to given set , then (1+i*0)* a = a *(1+i*0)1 = a.   1   is called the identity element of complex multiplication because multiplying it to any number returns the same number.

Inverse property:

For every number a, there is an another number b such that a * b = b* a =1. The number b is called the inverse element   of the integer a and is denoted (1/a).

Given set is satisfy the 4 property(closer,associative,identity,inverse),

Therefore, Given set is Group.