Artin's theorem on induced characters
# 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 characters induced from all cyclic subgroups of the group.

There is a similar but somehow more precise 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 Serre states in Chapter 9.2, 17 the theorem in the following, more general way:

Let $G$ finite group, $X$ family of subgroups.

Then the following are equivalent:

* $G=\cup _{g\in G,H\in X}g^{-1}Hg$
* $\forall \chi {\text{ character of }}G\exists \chi _{H},H\in X,d\in \mathbb {N} :d\chi =\sum _{H\in X}Ind_{H}^{G}(\chi _{H})$

This in turn implies the general statement, by choosing $X$ as all cyclic subgroups of $G$.

