Maschke's theorem averages an arbitrary linear projection into an invariant one and is the central linear-algebra form of coprime action. [@isaacs2008]
Definition
An elementary abelian \(p\)-group is an \(\mathbf F_p\)-space, and the condition becomes \(p\nmid|K|\).
Core result
Choose any projection \(\pi:V\to U\) and average it: \(\bar\pi=|K|^{-1}\sum_{k\in K}k^{-1}\pi k\). It remains a projection onto \(U\) and commutes with \(K\); its kernel is the invariant complement.
Structural properties
- Averaging requires the group order to be invertible in the field.
- Complete reducibility says every short exact sequence of modules splits.
- The coprime decomposition \(V=C_V(K)\oplus[V,K]\) is a special case.
- The source uses a group version for elementary abelian \(p\)-groups in the metacyclic transfer argument.
- The field need not be algebraically closed.
Maschke averaging is the linear prototype for choosing invariant complements in Schur–Zassenhaus and general coprime action.
Example and boundary
Knowledge network
Coprime action generalizes averaging, elementary abelian groups supply the vector space, and metacyclic transfer uses invariant complements.
Proof and sources
The group-action form and elementary abelian corollary are Theorems 10.16-10.17. [@isaacs2008]
