How big is the center of an orthogonal group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:44:25Z http://mathoverflow.net/feeds/question/32164 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32164/how-big-is-the-center-of-an-orthogonal-group How big is the center of an orthogonal group? soulphysics 2010-07-16T12:56:07Z 2010-07-16T12:56:07Z <p>How big is the center of an arbitrary orthogonal group $O(m,n)$?</p> <p>In the special case of the "circle group" $O(2)$, it's clear that $|\zeta O(2)|$ = 1. In the case of $O(3)$, it seems clear that the center has two elements $\zeta O(3) = \lbrace 1, -1 \rbrace$. I can see this by visualizing a sphere in an arbitrary $(i, j, k)$ basis, and observing that both the identity and the "complete" reversal $(i, j, k) \mapsto (-i, -j, -k)$ commute with everthing.</p> <p>But I'd like a simple way to see how the situation changes for more general orthogonal groups like the (inhomogeneous) Lorentz group $O(3,1)$.</p>