Findng a group between two groups

[anugrah]

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?

 

[n_bourbaki]

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

 

[morphism]

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

 

[anugrah]

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.

 

[d_leet]

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.

 

[n_bourbaki]

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.