A Structure Theorem for Plesken Lie Algebras over Finite Fields

W. Plesken found a simple but interesting construction of a Lie algebra from a finite group. Cohen and Taylor posed themselves the question of what the Plesken Lie algebra, which is the Lie subalgebra of the group algebra k[G] generated by the elements g − g−1, could be. The result is very fascinating: It turns out that the Lie algebra decomposition of the Plesken Lie algebra into simple Lie algebras corresponds to the irreducible characters of the group, more preciselyto the types and sizes of characters. The main idea of the present project was to find a similar decomposition result over a finite field instead of the complex numbers. When we lose algebraic closure, a lot of theorems do not hold anymore, and the problem becomes a lot harder. We take two approaches of reducing the Plesken Lie algebra modulo a prime: On one hand, we use the standard Chevalley basis approach, i.e., we find a Chevalley basis which by definition has integral coeffiecients, consider the Z lattice it defines and reduce it modulo a prime p. On the other hand, we start with the group algebra Fp[G] and define the plesken Lie subalgera directly. By comparing the results gotten from the different approaches, we get a fascinating theorem that relates the differences of the Plesken Lie algebra over a finite field of characteristic p to the way in which the prime p behaves in the splitting field of the group G, which is an extension of Q.