Question

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. Determine Spank(G) and its Jacobson radical.

Answer 1

We try to determine $S := \operatorname{Span}_k(G)$ . Using $\{e1 - e2, e1\}$ as a basis for $V$ , we have

$\mathcal{Q}(123) = \left( \begin{array}{cc}1 & -1\\ 0 & 1 \end{array} \right)$ and $\mathcal{Q}(12) = \left( \begin{array}{cc} - 1 & -1\\ 0 & 1 \end{array} \right),$

so $S \subseteq T$ , the $k$ -subalgebra of all upper triangular matrices in $\mathbf{M}_2(k)$ . Since $S$ is noncommutative, we must have $S = T$ . It follows that $\operatorname{rad} S = \operatorname{rad} T = k \cdot \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = k \cdot (1 - \mathcal{Q}(123))$ .