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?
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?
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.
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).