Restriction of a character (${\mathrm{Res}}_H^G\chi$)

For a subgroup $H\le G$, the restriction of a $G$-character $\chi$ to $H$ is the $H$-character obtained by evaluating the same function only on elements of $H$.

If $\chi$ is an irreducible character of a finite group $G$ and $H\le G$, thenEquality holds exactly when $\chi$ vanishes on $G\setminus H$.

Let $H◃G$ have index two, and let $\lambda$ be the nontrivial linear character of $G/H$ inflated to $G$. For an irreducible character $\chi$ of $G$, either ${\mathrm{Res}}_H^G\chi$ is irreducible and $\chi \ne \chi \lambda$, or the restriction is a sum of two distinct irreducible characters and $\chi =\chi \lambda$.