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Show that the group of positive rational numbers under

Chapter 4, Problem 37E

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QUESTION:

Problem 37E

Show that the group of positive rational numbers under multiplication is not cyclic.

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QUESTION:

Problem 37E

Show that the group of positive rational numbers under multiplication is not cyclic.

ANSWER:

Step 1 of 2

To prove that the group of positive rational numbers under multiplication is not cyclic.

If possible, let us assume that the group of positive rational number is cyclic.

Then, must have a generator other than the multiplicative identity.

Suppose, be a generator of the group such that . Then, every element of is some integer power of  i.e.,

                                                               

 

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