In a recent question on MSE I asked about conditions under which the canonical morphism $Out(G) \to Out(k[G])$ is injective.
Is it true that this morphism is indeed injective if $G$ is finite and $k=\mathbb{Z}$ ?
What I already know:
- It's not injective if $k$ is a field in non-modular characteristic. The kernel is $Out_c(G)$, the group of conjugacy class preserving automorphisms. That is Captain Lama's answer to my question.
- It is true if $k=\mathbb{Z}$ and $G$ is nilpotent. That's my own answer to the question.
- In the mean time I have also found out that the morphism is injective if $G$ is rational, i.e. if $g$ is cojugated to $g^k$ for every $g\in G$ and every $k\in\mathbb{Z}$ with $gcd(k,ord(g))=1$, e.g. $G$ a symmetric group.
Proof: If $\alpha\in Aut(G)$ is in the kernel of the morphism, say $\alpha(g)=ugu^{-1}$ for some $u\in\mathbb{Z}G^\times$, then $$u^\ast u g=u^\ast \alpha(g) u = u^\ast \alpha(g^{-1})^{-1} u = (u^\ast \alpha(g^{-1}) u)^\ast = (u^\ast u g^{-1})^\ast = g u^\ast u$$ where $^\ast$ is the antiautomorphism of the group algebra with $g\mapsto g^{-1}$. Hence $u^\ast u \in\mathbb{Z}G$. We want to prove that $u^\ast u = 1$.
Consider all the complex characters $\chi\in Irr(G)$ and their central characters $\omega_\chi: Z(\mathbb{C}G) \to \mathbb{C}$. We can pick a matrix representations $\rho_\chi: G\to GL_n(\mathbb{C})$ affording $\chi$ with $\rho(G)\subseteq U_n(\mathbb{C})$ so that $u^\ast u$ is mapped to some self-adjoint and positive definite matrix so that $\omega_\chi(u^\ast u)\in\mathbb{R}_{>0}$. Furthermore $u^\ast u\in Z(\mathbb{Z}G)$ is integral over $\mathbb{Z}$ so that $\omega_\chi(u^\ast u)$ is an algebraic integer. Moreover it must be in $\omega_\chi(Z(\mathbb{Q}G)) =\mathbb{Q}(\chi)$. Now if $G$ is rational, then all characters have rational values so that $\omega_\chi(u^\ast u)$ is a real, positive, rational, invertible integer. In other words $\omega_\chi(u^\ast u) = 1$, i.e $\rho_\chi(u^\ast u) = 1_{n\times n}$. Since $\chi$ was arbitrary, $u^\ast u=1$.
The only units of $\mathbb{Z}G$ with $u^\ast u=1$ are elements of the form $\pm g$ so that $\alpha\in Inn(G)$ as we wanted. QED.
Note that we can get the same conclusion under weaker conditions. For example if $\mathbb{Q}(\chi)=\mathbb{Q}(i)$, then the only real, positive, integral unit is also 1.
