Findng a group between two groups

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Discussion Overview

The discussion revolves around the existence of a group that lies between the set of rational numbers and the set of real numbers, specifically under the operation of multiplication. Participants explore whether such a subgroup exists and provide examples or counterexamples to support their claims.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that there exists uncountably many groups between the rationals and reals, assuming addition as the group law.
  • Another participant emphasizes that the reals form a 'very large' vector space over the rationals, suggesting a broader context for the discussion.
  • A later post clarifies that the group law in question is multiplication, and requests examples of subgroups under this operation.
  • One participant proposes a specific set, Q'={qkn | q is a nonzero rational number, and n is an integer}, as a group under multiplication, claiming it is a proper subgroup of the reals.
  • Another participant challenges the notion that the real numbers under multiplication form a group, indicating a potential misunderstanding or miscommunication about the group properties.
  • There is a suggestion for participants to work through the problem independently, indicating a shift towards self-exploration in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the existence of groups between the rationals and reals, particularly under multiplication. The discussion remains unresolved with multiple competing perspectives on the nature of these groups.

Contextual Notes

There are limitations regarding the definitions of groups under different operations, and the assumptions about the properties of real numbers under multiplication are not fully explored.

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