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A5

Part of 2010 Putnam

Problems(1)

Putnam 2010 A5

Source:

12/6/2010
Let GG be a group, with operation ∗*. Suppose that
(i) GG is a subset of R3\mathbb{R}^3 (but ∗* need not be related to addition of vectors);
(ii) For each a,b∈G,\mathbf{a},\mathbf{b}\in G, either a×b=a∗b\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b} or a×b=0\mathbf{a}\times\mathbf{b}=\mathbf{0} (or both), where ×\times is the usual cross product in R3.\mathbb{R}^3.
Prove that a×b=0\mathbf{a}\times\mathbf{b}=\mathbf{0} for all a,b∈G.\mathbf{a},\mathbf{b}\in G.
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