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In my last question I asked for examples of groups formed by real numbers where the operation is something different from addition or multiplication. With these words I think I could not convey what I wanted. In an attempt for further clarity in conveying my query I state the question as follow

" Are there examples of groups formed by real numbers where the binary operation of the group does not involve any addition or multiplication" I hope this time I will be getting appropriate answers.

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    Do you need the elements of the group to be all real numbers, or just a subset?2011-09-01
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    Also, what does "involves" mean?2011-09-01
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    @charles - Yes I want groups with the mentioned property whose elements are all real numbers.2011-09-01
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    It's not an answer, but one can show that any $C^1$ group structure on $\mathbb R$ (that is, a group structure whose multiplication $*$ and inverse are continuously differentiable functions) is in fact the addition in disguise (precisely: there is a $C^1$-diffeomorphism $f$ of $\mathbb R$ such that $f(x*y) = f(x) + f(y)$.2011-09-01
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    @charles- I meant the binary operation should not be multiplication.2011-09-01
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    For example, a+b, or a.b or a+b+(a.b), in a nutshell there should not be any addition or multiplication .2011-09-01
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    God forbid you get *inappropriate* answers...2011-09-01
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    Given an operation, how can I determine if it involves addition or multiplication?2011-09-01
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    Just to insist on The Chaz's irony, I find your opening of three questions on MSe/MO and the comments “I hope this time I will be getting appropriate answers” (here) and ”I hope in this site I may get better answers as compared to its stack exchange counterpart” (in MO) insulting.2011-09-01
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    **HINT** $\ $ Take any group $\:G\:$ with the same cardinality as $\:\mathbb R\:$ and transport the group structure of $\:G\:$ to $\:\mathbb R\:$ along any bijection of their underlying sets.2011-09-01
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    You *do* know that $\mathbb{R}$ is by itself just a continuum, right?2011-09-01
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    Rather bad form to open a second question with **identical** title and which is nothing but an attempt at explaining what you meant with the original. Much better would have been to **edit** your old question to add your clarification. As it is, I have voted to close [the old question](http://math.stackexchange.com/questions/60759/groups-of-real-numbers) as a duplicate, and invite you to not do this again in the future.2011-09-01
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    @PseudoNeo do you have a source for your statement about $C_1$ group operations on $\Bbb R$?2016-06-20
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    @MarioCarneiro: Well, I've learnt about it in a French exercise book on differential calculus (Rouvière, *Petit guide de calcul différentiel*). After your question I googled a bit and found this Conrad blurb: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/relativity.pdf which may suit you. (These blurbs are so amazing that one of these days I will launch a kickstarter project to make a bronze statue of K. Conrad).2016-07-05

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