Modular representation theory

Remarks

Example: $G = \Z/p\Z$ with generator $g$, $K = \F_p$, and $\rho(g) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. This is a representation because $\rho(g)^p = \begin{pmatrix} 1 & p \\ 0 & 1 \end{pmatrix} = I$ in characteristic $p$. The line $K e_1$ is invariant, but any other line $K(a e_1 + e_2)$ is sent to $K((a+1) e_1 + e_2)$, a different line, so $V$ is not a direct sum of invariant lines. Non-example: representations of $G$ over $\C$, or over a finite field whose characteristic does not divide $\abs{G}$, are ordinary representations; there Maschke's theorem makes every representation semisimple. Let $K$ be a splitting field for $G$. Brauer showed that the number of simple $KG$-modules is the number of conjugacy classes of elements of $G$ of order prime to $p$.