Orthogonal Groups O{script}(n) over GF(2) as Automorphisms

Authors：Kwak Y J^*

Source：Communications in Algebra, 2014, 42(5): 2099-2106.

DOI：10.1080/00927872.2012.755683

• Matrix rank

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group O{script}(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is O{script}(V, I) = O{script}(n). There is a nice result that O{script}(n) ≅ Aut(o{fraktur}(n)) over ℝ or ℂ, where o{fraktur}(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is "Yes" if it is GF(2). We show it explicitly with the combinatorial basis C{black-letter}. This is a verification of Steinberg's main result in 1961, that is, Aut(o{fraktur}(n)) is simple over the square field, with a nonsimple exception Aut(o{fraktur}(5)) ≅ O{script}(5) ≅ S{fraktur}6.

• Institution
  university of Colorado