MathDB

Let

G

be a group, with operation

*

. Suppose that
(i)

G

is a subset of

\mathbb{R}^3

(but

*

need not be related to addition of vectors);
(ii) For each

\mathbf{a},\mathbf{b}\in G,

either

\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}

\mathbf{a}\times\mathbf{b}=\mathbf{0}

(or both), where

\times

is the usual cross product in

\mathbb{R}^3.

Prove that

\mathbf{a}\times\mathbf{b}=\mathbf{0}

for all

\mathbf{a},\mathbf{b}\in G.

Problem Statement