Euler's theorem

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