The group (C, *)

The group (C, *) refers to the set of complex numbers C with the operation of multiplication (*). In this group, the identity element is 1, and each complex number has an inverse, except for the identity element itself.

Finite order elements

Finite order elements in a group are elements that, when raised to a certain power, result in the identity element. In the case of the group (C, *), a complex number z is a finite order element if there exists a positive integer n such that z^n = 1.

Calculating the finite order elements

To determine the total number of finite order elements in the group (C, *), we need to consider the possible values of n, the exponent, and count the distinct complex numbers that satisfy the equation z^n = 1.

Distinct values for n

Since we are looking for finite order elements, the exponent n should be a positive integer. We can consider the values of n starting from 1 and continue until we find a repeating pattern.

Case 1: n = 1
For n = 1, any complex number raised to the power of 1 will result in itself. Therefore, the only complex number satisfying z^1 = 1 is the identity element 1.

Case 2: n = 2
For n = 2, we need to find complex numbers z such that z^2 = 1. This equation can be rewritten as z^2 - 1 = 0, which factors into (z - 1)(z + 1) = 0. From this equation, we can see that the solutions are z = 1 and z = -1.

Case 3: n = 3
For n = 3, we need to find complex numbers z such that z^3 = 1. This equation can be rewritten as z^3 - 1 = 0, which factors into (z - 1)(z^2 + z + 1) = 0. The quadratic term z^2 + z + 1 does not have any real roots, so the only solution is z = 1.

Case 4: n = 4
For n = 4, we need to find complex numbers z such that z^4 = 1. This equation can be rewritten as z^4 - 1 = 0, which factors into (z - 1)(z + 1)(z^2 + 1) = 0. Again, the quadratic term z^2 + 1 does not have any real roots, so we have z = 1 and z = -1 as solutions.

Case 5: n = 5
For n = 5, we need to find complex numbers z such that z^5 = 1. This equation can be rewritten as z^5 - 1 = 0, which factors into (z - 1)(z^4 + z^3 + z^2 + z + 1) = 0. The quartic term z^4 + z^3 + z^2 + z + 1 does not have any real roots, so the only solution is z = 1.