Generating Set for Og over \$\mathbb Z_2\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:47:14Z http://mathoverflow.net/feeds/question/71154 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71154/generating-set-for-og-over-mathbb-z-2 Generating Set for Og over \$\mathbb Z_2\$ Larry 2011-07-24T21:30:20Z 2011-07-24T21:30:20Z <p>Hi, All:</p> <p>I am reading a claim that Og , the orthogonal group associated with a finite-dimensional vector space V over \$\mathbb Z_2\$ , and a quadratic form q defined therein , i.e., the group of linear transformations T:V-->V with q(v)=q(T(v)), is finitely-generated (and its generators are known) , because of the Cartan-Dieudonne' theorem. Now, the Cartan-Dieudonne theorem states that if V is a vector space of dimension n, then n is generated by <em>at most</em> n reflections. Now, my confusion here is that , since we are working over \$\mathbb Z_2\$ , orthogonal transformations are also symplectic, and the symplectic group is known to be finitely-generated by transvections (the generalization to vector spaces of shear maps in \$\mathbb R^n\$). Anyway, I am trying to figure out a generating set for Og as defined. Any Ideas? Thanks.</p>