Is the following action of a group on the character group of an Abelian group well-defined?

Question

Suppose $H$ is a finite group and $N$ is an abelian finite group and suppose $H$ acts by automorphisms of $N$, i.e. there is a group homomorphism $\phi:H \times N\to N$ such that $\phi_h\in\text{Aut}(N)$ for each $h\in H$. Then I want to show that the action of $H$ on the character group $\widehat{N}$ defined by $(h\cdot\chi)(n)=\chi(\phi_{h^{-1}}(n))$ is indeed a group action, but how do I know that $\chi\circ\phi_{h^{-1}}$ is a character for each $h\in H$?

Answer 1

You have to check three things: that your action is well defined, that $1 \cdot \chi = \chi$ for every $\chi \in \hat{N}$ and that $(gh) \cdot \chi = g \cdot (h \cdot \chi)$ for every $\chi \in \hat{N}$ and for every $g,h \in H$.

Recall that the a character $\chi$ of $N$ is a homomorphism from $G$ to $\mathbb{C}$. Since for every $h \in H, \phi_{h^{-1}}$ is an automorphism (in parciular, an endomorphism) from $N$ to $N$, then $\chi \circ \phi_{h^{-1}}$ is also an endomorphism from $N$ to $\mathbb{C}$, so it is a character.

The other two properties can be checked in a similar way, evaluating the characters in every element of $N$.