Artin's theorem on induced characters
Appearance
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
[edit]In Linear Representation of Finite Groups Serre states in Chapter 9.2, 17 [1] the theorem in the following, more general way:
Let finite group, family of subgroups.
Then the following are equivalent:
This in turn implies the general statement, by choosing as all cyclic subgroups of .
Proof
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References
[edit]- ^ Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. New York, NY: Springer New York. ISBN 978-1-4684-9458-7. OCLC 853264255.