For a finite group \(G\) and a field \(k\), describe the group algebra \(k[G]\). What is special about \(k=\CC\), and what changes when \(\operatorname{char}k\) divides \(|G|\)?
Solution
The group algebra \(k[G]\) is the \(k\)-vector space with basis \(G\) and multiplication obtained by extending the group law \(k\)-bilinearly: \[ \left(\sum_g a_g g\right)\left(\sum_h b_h h\right) =\sum_{g,h}a_gb_h(gh). \] A representation of \(G\) on a \(k\)-vector space is equivalently a left \(k[G]\)-module.
Nothing in the definition requires \(k=\CC\). The special feature of characteristic not dividing \(|G|\) is Maschke’s theorem: \[ \operatorname{char}k\nmid |G| \quad\Longrightarrow\quad k[G]\text{ is semisimple}. \] Equivalently, every finite-dimensional \(k\)-representation is completely reducible. The proof averages a projection over \(G\) and divides by \(|G|\).
Over \(\CC\) one also has a splitting field for every finite group: every irreducible complex representation is absolutely irreducible, and \[ \CC[G]\cong\prod_i M_{n_i}(\CC) \] by Artin–Wedderburn.
If \(\operatorname{char}k=p\) divides \(|G|\), the averaging argument cannot divide by \(|G|\), and Maschke’s theorem fails: \(k[G]\) is not semisimple. For example, if \(C_p=\langle g\rangle\), then in characteristic \(p\), \[ (g-1)^p=g^p-1=0, \] so \(k[C_p]\) contains the nonzero nilpotent element \(g-1\) and is not semisimple.