Artin's induction theorem

This article states an induction theorem: a result relating the linear characters and linear representations of a group with the characters/representations induced from the linear characters/representations of subgroups
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This fact is related to: linear representation theory
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Statement

Let $G$ be a finite group and $X$ a family of subgroups of $G$. Then the following are equivalent:

• The union of conjugates of elements of $X$ cover the whole of $G$
• Every character of $G$ is a rational linear combination of characters induced from characters of members of $X$

Proof

Simplifying the result to complex linear combinations

With a little linear algebra, we can show that if a character of $G$ is a complex linear combination of characters induced from members of $X$, all the coefficients are in fact rational. Thus, the problem reduces to showing that the class functions induced from members of $X$ span the space of all class functions on $G$.

This proof follows by using Frobenius reciprocity, and the fact that the only class function on $G$ which restricts to the zero function on every member of $X$, is the zero function on the whole of $G$.

Details

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