= Modular representation theory
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<Modular representation theory> studies <representations> of a finite <group> $G$ over a <field> $k$ whose <characteristic> divides $|G|$. In that case the averaging argument of <Maschke's theorem> cannot divide by $|G|$, so invariant subspaces need not have invariant complements.
For example, a generator of $C_p$ can act over $\mathbb F_p$ by $J=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$, since $J^p=I$. The first coordinate line is invariant but has no invariant complementary line. This forces the study of <indecomposable modules>, <projective covers>, radicals, and blocks, beyond the irreducible decomposition sufficient in <characteristic zero>.
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