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Definition of a group with redundancy?

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    I would define a group as follows:

    A group consists of a non-empty set G together with a binary operation (say, *) on G such that the following axioms hold:


    2. Relevant equations

    3. The attempt at a solution

    I know this is a trivial question, but doesn't a binary operation on a set (here G) necesserily imply that the set is closed under that binary operation? The only reason i ask is that many of the definitions of groups I have come across include both that * is a binary operation under which G is closed. Isn't this redundant?
    Last edited: Sep 28, 2011
  2. jcsd
  3. Sep 28, 2011 #2
    It all depends on how you define a binary operation. I would define it as a function

    [tex]*:G\times G\rightarrow G[/tex]

    If you define a binary operation like that, then the property you mention is indeed redundant.

    I guess, that many authors include that property because most readers aren't yet ready to view a binary operation as a function. So to make it easy on them, they include the axiom that G is closed.
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