How to prove that a set is a group? related difficult/challenge quesyion.

  1. "How to prove that a set is a group?" related difficult/challenge quesyion.

    1.In proving that a set is a group from definition, we have to show there is an inverse element for each element, do we have to show uniqueness of it?
    I believe this is unnecessary but people do this all the time.
    My argument:
    "For each [itex]a\in G[/itex], there exists a left inverse a' in G such that a'a=e." is enough.
     
  2. jcsd
  3. Re: "How to prove that a set is a group?" related difficult/challenge quesyion.

    Provided you have shown the multiplication is associative then the inverse has to be unique.
     
  4. Re: "How to prove that a set is a group?" related difficult/challenge quesyion.

    so in proving a set is a group it is redundant to show the uniqueness of inverse(just show existence is ok)?
     
  5. Fredrik

    Fredrik 10,301
    Staff Emeritus
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    Gold Member

    Re: "How to prove that a set is a group?" related difficult/challenge quesyion.

    It's only unnecessary if you have already proved that the axioms (xy)z=x(yz), ex=x and x-1x=e define a group. (The standard axioms are (xy)z=x(yz), ex=xe=x and x-1x=xx-1x=e). This is a bit tricky. Define a "left identity" to be an element e such that ex=x for all x, and a "left e-inverse" of x to be an element y such that yx=e. You can prove the theorem by proving all of the following, in this order.

    1. There's at most one right identity.
    2. If y is a left e-inverse to x, then x is a left e-inverse of y.
    3. e is a right identity (which by #1 must be unique).
    4. There's at most one left identity. (This means that e is an identity and is unique).
    5. Every left-e inverse is a right e-inverse. (Hint: Use 2).
    6. Every x has at most one left inverse. (This means that every element has a unique inverse).

    If you've done this once, or if this is a theorem in your book, then all you have to do to verify that the structure you're considering is a group is to verify that the alternative axioms stated above are satisfied.
     
  6. Landau

    Landau 905
    Science Advisor

    Re: "How to prove that a set is a group?" related difficult/challenge quesyion.

    It seems two different questions are asked:
    (1) Is it enough to show that each element has an inverse, instead of in addition showing this inverse is unique?
    (2) Is it enought to show that each element has a LEFT inverse, instead of showing that each element has both a right and a left inverse?

    Or is it

    (3) Is it enought to show that each element has a LEFT inverse, instead of showing that each element has both a right and a left inverse and that these are unique?
     
  7. Re: "How to prove that a set is a group?" related difficult/challenge quesyion.

    I use (2) as my argument for (1). I know that (2) is valid so (2) implys (1)?
     
  8. Fredrik

    Fredrik 10,301
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: "How to prove that a set is a group?" related difficult/challenge quesyion.

    Suppose that y and y' are left e-inverses of x, and z is a right e-inverse of x. Then y'x=yx, and if you multiply by z from the right, you get y'=y. Is that what you wanted to know? I'm not sure I understand what you're asking.

    By the way, in both #1 and #6, you're referring to a mere statement as an "argument". It's not an argument if it doesn't include some evidence for the claim.
     
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