Euler's theorem

This article gives the statement, and proof, of a particular subgroup in a group being conjugate-dense: in other words, every element of the group is conjugate to some element of the subgroup

Statement

Euler's theorem is any of the following equivalent statements:

• For any element of $SO(3,\mathbb {R} )$, there is an axis such that that element can be viewed as rotation about that axis
• Every element of $SO(3,\mathbb {R} )$ is conjugate to an element of $SO(2,\mathbb {R} )$ (where $SO(2,\mathbb {R} )$ is embedded as $2\times 2$-matrices)
• $SO(2,\mathbb {R} )$ is a conjugate-dense subgroup in $SO(3,\mathbb {R} )$