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:

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

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

Further reading

• http://www.math.toronto.edu/murnaghan/courses/mat445/artinbrauer.pdf
• https://math.stackexchange.com/questions/4854649/artins-theorem-for-the-linear-representation-of-finite-groups/4854696#4854696