Past exam of the algebra course of the University of Oxford / 2011 / 2 / b / ii / Solution

Choose bases $S_1=\mathbb{C}{s}_1$, $S_2=\mathbb{C}{s}_2$ and setBoth are simple modules. They are nonisomorphic because $e_{11}$ acts as the identity on $S_1$ and as zero on $S_2$. In the ascending composition series of a module above, the factors are, in order,The last factor is represented by $e_{22}$ modulo the span of $e_{11},e_{12}$.