# Is the Set a Group?

Decide whether each of the given sets is a group with respect to the indicated operation.

1- For a fixed positive integer n, the set of all complex numbers x such that x^n=1(that is, the set of all nth roots of 1),with operation multiplication.

2-The set of all complex numbers x that have absolute value 1, with operation multiplication. Recall that the absolute value of a complex number x written in the form x= a + bi, with a and b real, is given by lxl=l a+bi l=Sqrta^2+b^2

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#### Solution Preview

(1) this is a group

if x^n = 1 and y^n = 1, then certainly (xy)^n = 1, using the laws of exponents, so the set if closed under the group

operation

the group operation is also associative, since [(xy)z]^n= [x(yz)]^n = 1 where x^n = 1, y^n = 1, and z^n = 1

also, since x^n = 1, then take the inverse of both sides to get x^{-n} = 1, which ...

#### Solution Summary

Whether given sets are groups is determined. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.