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.