Findng a group between two groups

  • Thread starter anugrah
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  • #1
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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?
 

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  • #2
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There exist uncountably many. (We assume that you mean addition as the group law.)
 
  • #3
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.
 
  • #4
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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.
 
  • #5
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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.
 
  • #6
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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.
 

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