Fine Art


Maschke's theorem [1] is a theorem in group representation theory which concerns the decomposition of representations of a finite group into irreducible pieces. If (V, ρ) is a finite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariant subspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, the representation (V, ρ) is completely reducible. More generally, the theorem holds for fields of positive characteristic p, such as the finite fields, if the prime p doesn't divide the order of G.

It is named after Heinrich Maschke.[2]

Reformulation and the meaning

One of the approaches to representations of finite groups is through module theory. Representations of a group G are replaced by modules over its group algebra KG. Irreducible representations correspond to simple modules. Maschke's theorem addresses the question: is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

Let G be a finite group and K a field whose characteristic does not divide the order of G. Then KG, the group algebra of G, is a semisimple algebra.[3][4]

The importance of this result stems from the well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When K is the field of complex numbers, this shows that the algebra KG is a product of several copies of complex matrix algebras, one for each irreducible representation.[5] If the field K has characteristic zero, but is not algebraically closed, for example, K is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra KG is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.[6]

Returning to representation theory, Maschke's theorem and its module-theoretic version allow one to make general conclusions about representations of a finite group G without actually computing them. They reduce the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.


1. ^ Heinrich Maschke (July 1899) "Beweiss des Satzes, dass diejenigen endlichen linearen Substitutionesgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind" [Proof of the theorem that those finite linear substitution groups in which some everywhere vanishing coefficients appear, are intransitive], Mathematische Annalen, vol. 52, no. 3, pages 363-368.
2. ^ O'Connor, John J.; Robertson, Edmund F., "Maschke's theorem", MacTutor History of Mathematics archive, University of St Andrews, .
3. ^ It follows that every module over KG is a semisimple module.
4. ^ The converse statement also holds: if the characteristic of the field divides the order of the group (the modular case), then the group algebra is not semisimple.
5. ^ The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
6. ^ One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.


* Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4
* Serre, Jean-Pierre (1971). Linear representations of finite groups. Graduate texts in mathematics, 42. Springer-Verlag. ISBN 0-387-90190-6.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Hellenica World - Scientific Library