   Chapter 3.1, Problem 8E

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# In Exercises 1 − 14 , decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold.For a fixed positive integer n , the set of all complex numbers x such that x n = 1 (that is, the set of all n t h roots of 1 ), with operation multiplication.

To determine

Whether for a fixed positive integer n, the set of all complex numbers x such that xn=1(that is, the set of all nth roots of 1) is a group with operation multiplication.

Explanation

Given information:

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

Explanation:

Let us check if the set of all complex numbers x such that xn=1 with operation multiplication is a group or not by using the definition given below:

Suppose the binary operation is defined for element of a set G. The set G is a group with respect to , provided the following conditions hold:

1. G is closed under . That is xG and yG imply that xy is in G.

2. is associative. For all x,y,z in G, x(yz)=(xy)z.

3. G has an identity element e. There is an e in G such that xe=ex=x for all xG.

4. G contains inverses. For each aG, there exists bG such that ab=ba=e.

Let S={x|xn=1,x} for fixed positive integer n.

First condition:

Let x,yS such that x,y and xn=1,yn=1.

Then, xy and (xy)n=xnyn=1×1=1.

Thus, xyS and S is closed under multiplication

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