Can unit vectors $i, j, $ and $k$ be elements of a group?

Question

A collection of some vectors under group operations may form an abelian group. By the cancellation law of groups, the unit vectors $i, j, $ and $k$ along x-axis, y-axis and z- axis, respectively cannot be the elements of a group, because $i.j=i.k$ but $j\neq k$ implies that cancellation law fails for these vectors . Is these my intuitions correct? Please help me. Thanks in advance.

Answer 1

I suggest you look up 'quaternion group'.

Answer 2

The unit vectors $i,j,k$ can all be elements of the same abelian group. One specific example is if you take your set to be all vectors in $\mathbb{R}^3$ and use the operation of vector addition.