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Help Checking Some Proofs

  1. Feb 3, 2010 #1
    I have two proofs that I am uneasy about and one I'm having trouble with so hopefully I can figure out where I'm going wrong if I am. Ignore the weird numbers, its to help me organize the problems.

    14) Let G be a group with the following property: Whenever a, b and c belong to G and ab=ca, then b=c. Prove that G is Abelian. ("cross cancellation" implies commutativity.)

    [tex]a,b,c \in G[/tex]

    [tex]ab=ca[/tex] implies [tex]b=c[/tex]

    [tex]ab=ca[/tex]

    [tex]b=c[/tex]
    replace c with b in [tex]ab=ca[/tex] and

    [tex]ab=ba[/tex]

    which implies an Abelian group


    The problem I have with this is that I used cancellation and from some reason I get the feeling I'm not supposed to due to the parenthesis. The question seems to answer itself wit that statement because what they give in the question seems to imply cancellation. Unless Abelian means more than just commutative..



    26) Prove that if [tex](ab)^{2}=a^{2}b^{2}[/tex] in a group G, then ab=ba.

    Here's what I have:
    [tex](ab)^{2}=a^{2}b^{2}[/tex]

    [tex]a^{2}b^{2}=a^{2}b^{2}[/tex]

    [tex]aabb=aabb[/tex]

    [tex]a=a[/tex]

    Implies either:
    [tex]ab=ab[/tex] or [tex]ba=ba[/tex]

    [tex]a=\frac{ab}{b}[/tex] [tex]a=\frac{ba}{b}[/tex]

    [tex]\frac{ab}{b}=\frac{ba}{b}[/tex]

    [tex]b\frac{ab}{b}=b\frac{ba}{b}[/tex]

    [tex]ab=ba[/tex]


    15) Let a and b be elements of an Abelian group and let n be any integer. Show that [tex](ab)^{n}=a^{n}b^{n}[/tex]. Is this also true for non-abelian groups?

    I feel like I can use a bit from the above example, by starting off from
    here:
    [tex](ab)^{n}=a^{n}b^{n}[/tex]

    [tex]\overbrace{ab\cdot\cdot\cdot ab}^{n}=\overbrace{a\cdot\cdot\cdot a}^{n}\overbrace{b\cdot\cdot\cdot b}^{n}[/tex]

    but I feel like I'm working backwards from this to showing it is abelian instead of going from an abelian group to this which seems like what I should be doing...
     
    Last edited: Feb 3, 2010
  2. jcsd
  3. Feb 3, 2010 #2

    Mark44

    Staff: Mentor

    What parenthesis? I don't see any parentheses in the above.
    (ab)2 means (ab)(ab). So you have (ab)(ab) = a2b2. a and b are elements of a group, so each has an inverse, right?
    The two lines below aren't worth saying. It's obviously true that a equals itself, and that ab and ba equal themselves.
    I think the approach you need to take is an induction proof. You proof with ... is more akin to "armwaving."
     
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