   Chapter 3.1, Problem 9E

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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.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 + b i , with a and b real, is given by | x | = | a + b i | = a 2 + b 2 .

To determine

Whether the set of all complex numbers x that have absolute value 1, is a group with operation multiplication.

Explanation

Given information:

The absolute value of a complex number x written in the form x=a+bi, with a and b being real, is given by |x|=|a+bi|=a2+b2.

Explanation:

Let us check if the set of all complex numbers x that have absolute value 1 with operation multiplication is a group or not by using 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||x|=1}.

First condition:

Let x,yS such that x,y and |x|=1,|y|=1.

As the multiplication of complex numbers is closed, xy.

Also, |xy|=|x||y|=1×1=1

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