View source for Artin's theorem on induced characters

In representation theory, a branch of mathematics, '''Artin's theorem''', introduced by E. Artin, states that a character on a finite group is a rational linear combination of [[Induced character|characters induced]] from all cyclic subgroups of the group.
There is a similar but somehow more precise [[Brauer's theorem on induced characters|theorem due to Brauer]], which says that the theorem remains true if "rational" and "cyclic subgroup" are replaced with "integer" and "elementary subgroup".

== Statement ==
In Linear Representation of Finite Groups [[Biography:Jean-Pierre Serre|Serre]] states in Chapter 9.2, 17 <ref>{{Cite book |last=Serre |first=Jean-Pierre |url=https://www.worldcat.org/oclc/853264255 |title=Linear Representations of Finite Groups |date=1977 |publisher=Springer New York |isbn=978-1-4684-9458-7 |location=New York, NY |oclc=853264255}}</ref> the theorem in the following, more general way:

Let <math>G</math> finite group, <math>X</math> family of subgroups.

Then the following are equivalent:

# <math>G = \cup_{g\in G, H \in X} g^{-1}Hg</math>
# <math>\forall \chi \text{ character of } G \exists \chi_H, H \in X, d \in \N : d \chi = \sum_{H\in X} Ind_H^G(\chi_H)</math>

This in turn implies the general statement, by choosing <math>X</math> as all [[Cyclic group|cyclic]] subgroups of <math>G</math>.

== Proof ==

== References ==
<references />

== Further reading ==
* http://www.math.toronto.edu/murnaghan/courses/mat445/artinbrauer.pdf



[[Category:Representation theory of finite groups]]

{{Sourceattribution|Artin's theorem on induced characters}}