# Counterexample in Serre about Artin's induction theorem with $\mathbb{Z}$ coefficients

After reading the exposition about Artin's induction theorem for $$\mathbb{Q}$$-representations, for instance in Serre's Linear Representations of Finite Groups, 12.5, prop 25, one can be tempted to make the following generalization:

Let $$G$$ be a finite group. Then $$G$$ satisfies property (A): Every character over $$\mathbb{Q}$$ of $$G$$ is a $$\mathbb{Z}$$-linear combination of characters induced from trivial characters of arbitrary subgroups.

In the following, I will say that a given $$\mathbb{Q}$$-character is nice if it is a $$\mathbb{Z}$$-linear combination of characters induced from trivial characters of arbitrary subgroups.

Serre immediately mentions that there is no induction theorem because the product $$H\times C_3$$ of the quaternion group and the cyclic group provides a counterexample. After some computations, this looks wrong to me; let me explain.

If $$C$$ is a cyclic group of order $$n$$, its irreducible $$\mathbb{Q}$$-representations are the $$\mathbb{Q}(\zeta_d)$$ for $$d$$ dividing $$n$$ and $$\zeta_d$$ a primitive $$d$$-th root of unity, where the generator acts by multiplication by $$\zeta_d$$. Since we can decompose the regular representation as the direct sum of these, by Möbius inversion we get that the irreducible $$\mathbb{Q}$$-characters are nice; so the same holds for every $$\mathbb{Q}$$-character.

If $$G=H\times K$$, we have $$\mathrm{ind}^{H'}_H(\phi)\cdot\mathrm{ind}^{K'}_K(\psi)=\mathrm{ind}^{H'\times K'}_{H\times K}(\phi\cdot\psi)$$ so property (A) holds for a product of groups if it holds for each factor. Therefore it holds for all abelian groups.

We are reduced to studying the irreducible $$\mathbb{Q}$$-characters of $$H=\{1,-1,i,-i,j,-j,k,-k\}$$; they are given by the trivial representation, the three sign representations of degree one with respective kernels $$$$, $$$$ and $$$$, and a faithful representation of degree 4. The four first representations come from a cyclic quotient of $$H$$ so they are nice; finally, since the regular representation is the direct sum of all irreducible representations each with multiplicity one, it follows that the last representation also has a nice character.

Is there something wrong in my reasoning ? Do we know of an actual counter-example ?

If $$G=H\times K$$, we have $$\mathrm{ind}^{H'}_H(\phi)\cdot\mathrm{ind}^{K'}_K(\psi)=\mathrm{ind}^{H'\times K'}_{H\times K}(\phi\cdot\psi)$$ so property (A) holds for a product of groups if it holds for each factor. Therefore it holds for all abelian groups.
The second part of that statement is wrong. I have wrongly assumed that the product of $$\mathbb{Q}$$-irreducible characters is $$\mathbb{Q}$$-irreducible. This can fail because the Schur index of a product only divides the product of Schur indexes. The example I talked about was in the old french edition; in the 1977 english edition, it is given with more details as exercise 13.4. Let me elaborate.
Let $$\phi$$ denote the faithful irreducible complex character of the quaternion group $$H$$ and $$\psi=2\phi$$ the faithful $$\mathbb{Q}$$-irreducible character of $$H$$ (see Linear representation theory of quaternion group). Let $$\chi$$ denote the character of the faithful irreducible rational representation of $$C_3$$ given by $$\mathbb{Q}(e^{2i\pi/3})$$. Then Serre explains how to construct a faithful $$\mathbb{Q}$$-irrreducible representation of $$H\times C_3$$ with character $$\phi\cdot\chi$$; in particular the character $$\psi\cdot\chi=2(\phi\cdot\chi)$$ is not $$\mathbb{Q}$$-irreducible.
Moreover, one computes $$<\phi\cdot\chi,\phi\cdot\chi>=2$$ and $$\phi\cdot\chi$$ is of degree 4, hence the multiplicity of $$\phi\cdot\chi$$ in the regular representation is $$\frac{<1_{1}^{H\times C_3},\phi\cdot\chi>}{<\phi\cdot\chi,\phi\cdot\chi>}=\frac{(\phi\cdot\chi)(1)}{2}=2$$ which implies by exercise 13.3 that $$\phi\cdot\chi$$ is not a $$\mathbb{Z}$$-linear combination of permutation characters.
For more information about which virtual $$\mathbb{Q}$$-characters are linear combinations of permutation characters, one can read Bartel, Dokchitser, Rational representations and permutation representations of finite groups