nLab Maschke's theorem

Statement

Theorem

(Maschke’s theorem) Given a finite group $G$ and a field $k$, then the following equivalent statements hold iff the characteristic of $k$ does not divide order ${\vert G \vert}$:

1. every $G$-linear representation is completely reducible in that it is a direct sum of irreducible representations,

   (this is, in modern paraphrase, the statement in the title of Maschke 1899)

2. the group algebra $k[G]$ is semisimple

If the base $k$ is just a commutative unital ring, then there is the following statement:

Theorem

If ${|G|}\cdot 1_k$ is invertible in $k$, then an exact sequence of $k[G]$-modules splits iff it splits after applying the forgetful functor from $k[G]$-modules to $k$-modules (and the splitting in ${}_{k[G]}Mod$ can be functorially constructed from the splitting in ${}_{k}Mod$).

If $k$ is a field, it follows that $k[G]$ is semisimple, so that Thm. can be understood as a generalization of Maschke’s theorem . This is also one of the motivations for the concept of separable functors.

The importance of the classical Maschke’s theorem is that much is known about the structure of semisimple rings (starting with, e.g., Wedderburn's theorem).

Related entries

• separable functor

Literature

Named after:

• H. Maschke: Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen, Math. Ann. 50 (1898) 492–498 [doi:10.1007/BF01444297]

• H. Maschke: Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind, Math. Ann. 52 (1899) 363–368 [doi:10.1007/BF01476165]

Textbook accounts:

• Jean-Pierre Serre; section 1, Thms. 1, 2 in: Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer (1977) [doi:10.1007/978-1-4684-9458-7, pdf]

  (stated just over the complex numbers)

• David S. Dummit, Richard M. Foote; §8.1 Thm. 1 (p. 849) in: Introduction to the Representation Theory of Finite Groups, part VI of: Abstract Algebra, Wiley (2003) [ISBN:978-0-471-43334-7, pdf]

See also:

• Wikipedia, Maschke’s theorem