# Why is the Poincaré group non-abelian?

Based on what I've learned, I gather the Poincaré group is the group of isometries of Minkowski spacetime and it is a non-abelian Lie group.

Why is it non-abelian?

Or perhaps rather, does the fact that it is non-abelian mean anything significant for the differences we see between pre-SR and SR theory?

• Do you know that already 3D spatial rotations are non-abelian? – Qmechanic Jan 30 '15 at 22:50
• The group of rotations of ordinary 3-space is non-abelian as well – doetoe Jan 30 '15 at 22:50
• @doetoe: Touche! – Qmechanic Jan 30 '15 at 22:51
• @Qmechanic Oh, is that related to the right-handed rule idea? I haven't really learned much about this subject formally from a geometrical perspective. But I'm planning to. – Stan Shunpike Jan 30 '15 at 23:13

Already the simplest isometry you encounter in physics, that of rotations in ordinary Euclidean three space plus (or, rather, semi-direct product with) translations, is non-abelian, since the group of rotations $\mathrm{SO}(3)$ alone is non-abelian - it is not the same to first rotate about one axis and then about another or to do so in reverse.