Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. What is its unipotent radical?
The unipotent radical of G isthe unique maximal normal unipotent subgroup. This unipotent radical consists of g ∈ G which acttrivially on the composition factors of kGV . In our case,this subgroup is clearly <(123)>.