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 [1] the theorem in the following, more general way:
Let [math]\displaystyle{ G }[/math] finite group, [math]\displaystyle{ X }[/math] family of subgroups.
Then the following are equivalent:
- [math]\displaystyle{ G = \cup_{g\in G, H \in X} g^{-1}Hg }[/math]
- [math]\displaystyle{ \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]\displaystyle{ X }[/math] as all cyclic subgroups of [math]\displaystyle{ G }[/math].
Proof
References
- ↑ Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. New York, NY: Springer New York. ISBN 978-1-4684-9458-7. OCLC 853264255. https://www.worldcat.org/oclc/853264255.
Further reading
![]() | Original source: https://en.wikipedia.org/wiki/Artin's theorem on induced characters.
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