Proof of the maximality of the orthogonal group in the unimodular group

Abstract

Let D be a real inner product space of dimension n. The inner product is assumed to be positive definite. The unimodular group U and the orthogonal group O of D are defined by $$ U = \left\{ {H\left| {H \in L,\left| {\det H\left| { = 1} \right.} \right.} \right.} \right\} $$ (1) and $$ O = \left\{ {Q\left| {Q \in L,{Q^{T}} = {Q^{{ - 1}}}} \right.} \right\} $$ (2) , where L is the algebra of all linear transformations of D into itself. It is the purpose of this note to give a simple proof of the fact that O is a maximal subgroup of U.