Brauer's induction theorem

Name
This result is also termed the characterization of characters lemma or characterization of linear characters lemma.

Version involving elementary subgroups and arbitrary characters
Let $$G$$ be a finite group. Brauer's induction theorem states that every character of $$G$$ is a combination, with integer coefficients, of characters induced from fact about::elementary subgroups of $$G$$. In other words, the character ring of $$G$$ (over integers, for representations over complex numbers) is generated as a $$\Z$$-module by characters induced from elementary subgroups.

Here, the term induced is used in the sense of fact about::induced class function. For a character of a representation on $$H$$, the induced class function on $$G$$ by the character is the same as the character of the fact about::induced representation from $$H$$ to $$G$$.

Version involving elementary subgroups and linear characters
Let $$G$$ be a finite group. This strong form of Brauer's induction theorem states that every character of $$G$$ is a combination, with (possibly negative) integer coefficients, of characters induced from fact about::linear characters (i.e., characters of one-dimensional representations) on elementary subgroups.

Version involving linear characters
Let $$G$$ be a finite group. Every character of $$G$$ is a combination, with (possibly negative) integer coefficients, of characters induced from linear characters (i.e., characters of one-dimensional representations) on subgroups.

Similar facts

 * Artin's induction theorem
 * Dress induction theorem

For a complete list of induction theorems, refer Category:Induction theorems.

Applications

 * Sufficiently large implies splitting

Facts used

 * 1) uses::Elementary implies nilpotent, uses::Nilpotent implies supersolvable
 * 2) uses::Finite supersolvable implies monomial
 * 3) uses::Elementariness is subgroup-closed

The first formulation implies the other two
By facts (1) and (2), every linear representation of an elementary subgroup is monomial. In particular, every irreducible linear representation is induced from a degree-one representation, i.e., from a linear character. Further, by fact (3), any subgroup of an elementary group is elementary. Thus, using the fact that induction of representations is transitive, we obtain that the $$\Z$$-span of representations induced from linear characters of elementary subgroups is the same as the $$\Z$$-span of representations induced from all characters of elementary subgroups.

This shows that the first formulation implies the second. The third formulation follows naturally from the second.

Proof of the first formulation
The idea is to show that for each $$p$$ dividing the order of the group, the constant function sending every element to the index of the $$p$$-Sylow subgroup, is a linear combination of characters induced from $$p$$-elementary subgroups. Then, we can combine these constant functions to obtain the constant function that sends everything to 1. After that, suitable tensoring gives the result that every character occurs as a linear combination of characters induced from elementary subgroups.