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Putnam 2010 A5


Let G be a group, with operation \star. Suppose that:

  1. G is a subset of \Bbb{R}^3;
  2. For each a,b \in G either a \times b=a\star b or a\times b=0 (or both), where \times is the usual cross product.

Prove that a\times b=0 for all a,b \in G.

Putnam A5

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  1. bekaarbokbok
    March 17, 2011 at 5:37 pm

    I have a partial solution.

    Let e be the identity element of G.
    If this is the only element in G, we are done.

    Suppose not.
    Then there are two cases:
    A) e is not 0
    B) e = 0

    In either case, choose any other element of G, say a.
    Now either
    1) a x e = a*e = a
    or
    2) a x e = 0

    In case A):
    If a is not zero, then a x e must be perpendicular to a.
    So, if a x e = a, then a = 0

    On the other hand, if a x e = 0, then a is some multiple of e

    This if e is not 0, then every other element of G must be some multiple of e
    (including 0).

    But then a x b = 0 for all elements in G

    The case e = 0 is giving me a lot more trouble.
    Working on it.

    • March 17, 2011 at 8:45 pm

      Thank you. I will try and post a solution too. I think the case e=0 is harder than the other one.

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