# Findng a group between two groups

we all know that set of rationals i a subgroup of set of reals. 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?

There exist uncountably many. (We assume that you mean addition as the group law.)

morphism
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It will help your intuition if you notice that the reals form a 'very large' vector space over the rationals.

sorry, forgot to mention that the group law is multiplication. I wanted everything with respect to multiplication. And if there exists a subgroup between these two groups (even with respect to addition ) please give an example of the same.

sorry, forgot to mention that the group law is multiplication. I wanted everything with respect to multiplication. And if there exists a subgroup between these two groups (even with respect to addition ) please give an example of the same.

Pick any irrational number k and consider the set Q'={qkn | q is a nonzero rational number, and n is an integer}

It is easy to show that this is a group under multiplication, and easy to show that the nonzero rationals are a subgroup of this group, but clearly since this group is countable it is proper subgroup of the reals.

sorry, forgot to mention that the group law is multiplication.

The real numbers under multiplication are not a group.

I wanted everything with respect to multiplication. And if there exists a subgroup between these two groups (even with respect to addition ) please give an example of the same.

It would be better if you tried to work it out yourself.