how do you prove that the set of positive real numbers under multiplication is not cyclic

in general, how do you prove that something is not cyclic?

thanks

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- May 28th 2010, 02:44 PMalexandrabel90cyclic
how do you prove that the set of positive real numbers under multiplication is not cyclic

in general, how do you prove that something is not cyclic?

thanks - May 28th 2010, 03:02 PMDrexel28
- May 29th 2010, 12:41 AMalexandrabel90
So if a subgroup is not countable, then it is not cyclic.

so in this case, if the element in the group has infinite order, then it is not cyclic..but im not sure how does that help in writing a proof for this question:(

in the function that you have defined, wont there be no identity under multiplication? im not sure how to define functions and always wonder how are they defined. - May 29th 2010, 02:07 AMHallsofIvy
No, infinite is not enough- it must be uncountable. The set of positive real numbers is uncountable, therefore any group having the set of positive real number as underlying set, in particular the "set of positive real numbers under multiplication" is not cyclic.

Quote:

in the function that you have defined, wont there be no identity under multiplication? im not sure how to define functions and always wonder how are they defined.

**is**the identity. Note that this is simply a "one-to-one" function from the non-negative integers to the group. It is not claimed to be an "isomorphism". - May 29th 2010, 03:43 AMalexandrabel90
- May 29th 2010, 06:28 AMroninpro
Usually the uncountability of the real numbers is a standard fact that you can use. However, if you need to prove it, see Wikipedia:

Cantor's diagonal argument - Wikipedia, the free encyclopedia