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$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them?

I started thinking about this question last night after looking at the structure of the cosets of $\R / \Q$ What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?. I did some searching on SO and google but didn't find anything giving a full categorization (or even a partial one) of the subgroups of $\R$.

Here are the subgroups that I came up with so far:

  • $\Z$ (there are no finite subgroups and $\Z$ is the universal smallest subgroup I think)
  • n$\Z$ eg 2$\Z$ all even numbers
  • a$\Z$ where a is any real number, including a in $\Q$ which "nest" nicely in each other
  • $\Z$[a] - group generated by adding one real a to $\Z$
  • n$\Z$[a] which equals $\Z$[na] and so is just a case of the one above
  • Dyadic rationals eg a numbers of the form a/2b or similar subgroups such as a/3b, a/2b7c etc
  • $\Q$
  • $\Q$[a]
  • $\Q$[a in A] where A is a subset of $\R$ - could be finite, countable or uncountable. Group generated by adding all elements of A to $\Q$ eg $\Q[\sqrt2]$

It is clear that the "n$\Z$ subgroups" n$\Z$ and m$\Z$ are related according to the gcd(n,m)

Also when H is a subgroup of R looking at the structure of the cosets of R / H. eg for H any of the Z subgroups we get R / H homomorphic to [0,1) or the circle. For H one of the Q subgroups it is more complex and I currently don't have ideas on the larger subgroup cosets

I am not clear how "big" a subgroup H can get before it becomes the whole of R. I do know that if it contains any interval then it is the whole of R. But what about H with dimension less than 1?

I am aware of one question on SO about the proper measurable subgroups of R having 0 measure Proper Measurable subgroups of $\mathbb R$, one on dense subgroups Subgroup of $\mathbb{R}$ either dense or has a least positive element? and one on the subgroups of Q How to find all subgroups of $(\mathbb{Q},+)$ but that is all my searching found so far.

Why is this question interesting? 1) there seem to be so many subgroups and they are related in many groupings 2) I think the subgroups related to the structure of the reals in some subtle ways 3) I know the result for the complete classification of all finite subgroups was a major result so wondering what has been done in this basic uncountable case.

If anyone has any insight, intuition, info, papers or theorems on subgroups of R and how they are interrelated that would be interesting.

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    ...and that is why I am not going to try and edit any question long enough ever again (well not ever, but for a while...)2012-06-02
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    Some of the subgroups of $\mathbb Q$ are missing from your list, so I'd go work through the answer of that one first (e.g. the dyadic rationals, off the top of my head)2012-06-02
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    This may be helpful http://mathoverflow.net/questions/59978/additive-subgroups-of-the-reals2012-06-02
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    @benmachine - thanks - I added in dyadic rationals and link to article on them2012-06-02
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    @Norbert thanks for that article - looks interesting, though I hope the task is not completely hopeless!2012-06-02
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    There are more like dyadic rationals (I mean, just think of $3^b$ on the denominator, or possibly $2^a3^b$, or ...) I didn't mean so much "you forgot the dyadics!" as "you forgot some subgroups of the rationals, maybe you should understand those fully first"2012-06-03
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    @benmachine - thanks for clarifying, I will look into those more and I found a proof of the classification of subgroups of Q at http://answers.yahoo.com/question/index?qid=20080425160119AA19f8E which shows that they are all either finitely generated or of the type of the dyadic rationals.2012-06-03
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    Oops, I meant "there are more subgroups that are similar to the dyadic rationals", rather than "there are more dyadic rationals". The original description you had of the dyadics as $a2^{-b}$ was correct.2012-06-03
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    I am thinking that the structure of the subgroups of R might be clearer if we look at the generators of them. For example Z = [1], nZ = [n], aZ = [a], Dyadic =[1/2^n n in N], Q = [1/p p prime] etc. Then it is clearer which subgroups are subgroups or supergroups of others. Perhaps this idea can be extended to infinite (uncountable?) sets of irrational generating elements...2012-06-03
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    @benmachine - thanks for correction of definition of dyadic rationals - I updated the question text to reflect this.2012-06-03

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