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A Question on Semantics Regarding Group Theory

  1. Feb 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Is the set of a single element {e} with the multiplication law ee = e a group?


    2. Relevant equations
    none.


    3. The attempt at a solution
    Yes, it is a group. But that is not my question. My question is how do you ask the question? If I were face to face with you and wanted to ask you the question, would I say, "Is the set of a single element e with the multiplication law e multiplied by ANOTHER e (another element in the group) equivalent to the identity element?"

    Also, if I am wrong about it being a group...who cares. If I get the semantics down first, I will better understand what the problem is asking.

    Thanks everyone!
     
  2. jcsd
  3. Feb 20, 2012 #2
    Hi Mindstein.
    I am not sure I get your question.
    {e} is a group (for say, multiplication) IF e is the neutral element (or identity).
    for any group whose neutral element is e, {e} is (with the group itself) the trivial subgroup.
    What do you mean 'another e' ?
    If you are thinking that you can take any element of the original group and take put it in a sngleton and wonder if this singleton is also a group, than the answer is no. it's only valid for e.
    for instance, take (N, +), 0 is its 'e', so {0} is a group, but {1} is not since 1+1 does not belong to {1}
    sorry if I didn't get your question

    cheers...
     
  4. Feb 20, 2012 #3

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi Mindstein! :smile:
    Yes.

    If you're worried that you can't pick two e's in S, it doesn't matter …

    the law about multiplication is defined on S x S, not on S itself (where S is the set),

    and (e,e) is an element of S x S :wink:
     
  5. Feb 20, 2012 #4
    Thanks tiny-tim and oli4, you all sure do know how to get a brother past his problems!
     
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