we all know that set of rationals is a subgroup of set of reals under the group law multiplication. my question is whether there exsts a group between these tw groups. f yes what it can be? and if no, how to prve the non-existence?

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- Jul 25th 2008, 01:09 AManugrahfindng a group between two groups...
we all know that set of rationals is a subgroup of set of reals under the group law multiplication. my question is whether there exsts a group between these tw groups. f yes what it can be? and if no, how to prve the non-existence?

- Jul 25th 2008, 01:36 AMCaptainBlack
- Jul 25th 2008, 11:34 AMJaneBennet
Do you mean set of nonzero rationals and set of nonzero reals under multiplication? If

*d*is any square-free positive integer (i.e. not a perfect square), the set*A*of all*a*+*b*√*d*, where*a*and*b*are rationals and not both 0, is a multiplicative group lying strictly between the multiplicative groups of the nonzero rationals and the nonzero reals. - Jul 25th 2008, 11:44 AMThePerfectHacker
There are many such groups.

Let \alpha = \sqrt[n]{p} for a prime $\displaystyle p$.

Then let $\displaystyle \mathbb{Q}$($\displaystyle \alpha$) = {a_0+a_1\alpha+...+a_{n-1}\alpha^{p-1} } where a_0,a_1,...,a_{n-1} are in Q.

Then this is a group under addition and multiplication (without zero).