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Groups of real numbers

A group as we all know is a set A with a binary operation "" defined on it such that for every a,b belongs to A ab belongs to A, the operation is assocaitive, there exists an identity element and for every a in A there exists a unique p in A such that a*p=identity element. Can anyone provide me examples of groups where the operation defined on the set is not addition + or multiplication x.

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    -1: The question admits infinitely many answers (in a very strong sense!) and is answered in literally every course in group theory (e.g., any non commutative group does the trick).2011-08-30
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    Should this question get made into a community wiki, or have I misunderstood the term?2011-08-30
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    @PseudoNeo: That doesn't make it any less valuable a question, because this person might not have the opportunity to attend a course in group theory. I really don't think we should be picky about which questions we think too 'standard' to ask. It's highly conceivable that they might be learning group theory from a non-standard (or just plain bad) textbook and not know it.2011-08-30
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    If one interprets your question as "what topological/Lie group structures exist on subspaces of $\mathbb{R}^n$", it becomes very interesting!2011-08-30
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    @Billy- So, you think I.N.Herstein's algebra text is a bad one?2011-08-31
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    @Billy: by "every course", I meant every material about group theory. Even if the OP hasn't access to teachers or books, there's still plenty of Internet material, starting with wikipedia.2011-08-31
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    @Primeczar: I don't know it.2011-08-31
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    @PseudoNeo: Right. But, again, how do we know that the OP isn't simply (completely inadvertently) using poor material? Wherever they're learning group theory from, if the only examples they've come across are examples of multiplication and addition, something's clearly gone wrong, and it's not their fault. It is no one's position here to judge others on the quality of the free online material (or book, or teacher) they happen to be learning group theory from. They possibly don't know any better, and that in itself is a great reason to ask.2011-08-31
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    Well, regardless of the more or less improbable circumstances of the OP's biography one can think of, I think that a group theory question answered in each of the first answers spat out by a "group theory" Google query isn't a great MSe question. My "-1" only says that: I haven't judged anyone, nor asked for the deletion of the question or the stoning of the OP... BTW, a group theory course mentioning only the addition and multiplication of numbers wouldn't be “non-standard”, “plain bad” or “poor”: it would be an atrocious piece of fæces.2011-08-31

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