Moufang plane and Spin groups

In this thesis we consider the action of the exceptional simple Lie group F4 on the so called (real) Moufang plane OP2 R. The goal of this thesis is to present a proof of the transitivity of this action, which is as complete as possible. We first define related concepts such as Clifford algebras, the groups Pin(r, s) and Spin(r, s) and the algebra of octonions O, and we prove their basic properties. The group F4 is defined as the automorphism group of the algebra J3(O) of hermitian octonionic matrices of order three. The Moufang plane is defined as a suitable subset of J3(O). In the group F4 we find isomorphic copies of the groups Spin(0, 8) and Spin(0, 9). By applying certain auxilliary results from the previous chapters we obtain the desired proof of the transitivity of the action of F4 on OP2 R.