# Artin's theorem and Brauer's theorem on Characters

Artin: Let $$\chi$$ be rational valued complex character of (finite) group $$G$$. Then $$\chi$$ can be written as $$\mathbb{Q}$$-linear combination of characters $$1_H^G$$ for some cyclic subgroups $$H$$ of $$G$$.

In the theorem of Brauer, the subgroups $$H$$ are allowed to be elementary subgroups it is quite general than Artin's theorem in following sense:

Brauer: Every irreducible complex character $$\chi$$ of $$G$$ can be written as $$\mathbb{Z}$$-linear combinations of characters $$\lambda_H^G$$ for some subgroups $$H$$ of $$G$$, which are elementary subgroups, and $$\lambda$$ is a linear character of $$H$$.

Q. In the theorem of Brauer, comparing with Artin,

1) subgroups $$H$$ are allowed to be elementary (which include cyclic subgroups of $$G$$ also).

2) For elementary subgroups $$H$$, we consider $$\lambda_H^G$$, with $$\lambda$$ a linear character of $$H$$, not necessarily $$1$$.

I was wondering, what will happen if we consider subgroups $$H$$ to be elementary but the characters of $$G$$ to be $$1_H^G$$ (instead of $$\lambda_H^G$$); these induced characters are rational valued; so by integral combination of such characters of $$G$$, can we get all the rational valued characters of $$G$$?

A (sub)group $$K$$ is said to be $$p$$-elementary, for a prime $$p$$, if $$K=C_m\times P$$ where $$C_m$$ is cyclic group of order coprime to $$p$$ (may be trivial) and $$P$$ is a $$p$$-group (may be trivial). We say that $$K$$ is elementary (sub)group if it is $$p$$-elementary for some prime $$p$$.

If we take the characters $$\{ (1_H)^G \,\, | \,\, H \mbox{ is elementary subgroup of G} \}$$, then an integer valued character of $$G$$ may not be integral combination of these specific induced characters.
Consider the quaternion group $$G=Q_8=\langle x,y\rangle$$. All subgroups of $$Q_8$$ are elementary subgroups: $$1, \,\, \langle x^2\rangle,\,\, \langle x\rangle,\,\, \langle y\rangle,\,\, \langle xy\rangle,\,\, G.$$ Consider trivial characters of these subgroups, and induce to $$G$$, and denote them, respectively by $$\psi_1, \ldots, \psi_6.$$
Consider $$\chi$$ the degree $$2$$ complex irreducible character of $$G$$, it is integer values.
Suppose $$\chi=a_1 \psi_1 + \cdots + a_6\psi_6$$ with $$a_i\in\mathbb{Z}$$. Then $$(*) \,\,\,\,\,\,\,2=\chi(1)=a_1 \psi_1(1)+\sum_{i>1} a_i\psi_i(1) \mbox{ and } -2=\chi(x^2)=a_1 \psi_1(x^2)+\sum_{i>1} a_i\psi_i(x^2)$$ Note for $$2\le i\le 6$$, $$\psi_i(1)=\psi_i(x^2)$$; whereas, $$\psi_1(1)=8$$ and $$\psi_1(x^2)=0$$.
Subtracting equations in (*) and noting that the five-term sums in both are same, we get $$4=8a_1$$, so $$a_1\notin \mathbb{Z}$$.