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Findng a group between two groups

  1. Jul 24, 2008 #1
    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?
     
  2. jcsd
  3. Jul 24, 2008 #2
    There exist uncountably many. (We assume that you mean addition as the group law.)
     
  4. Jul 24, 2008 #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.
     
  5. Jul 24, 2008 #4
    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.
     
  6. Jul 24, 2008 #5
    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.
     
  7. Jul 25, 2008 #6
    The real numbers under multiplication are not a group.


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