= Solution
Because $[G:H]=2$, part (a) says that the squared norm of $\chi\mathbin{\downarrow}_H$ is either one or two. It is one exactly when the restriction is irreducible, whereas it is two exactly when $\chi$ vanishes throughout $G\setminus H$. Thus condition (1) is equivalent to the existence in condition (2).
On $H$ we always have $\chi\lambda=\chi$, while outside $H$ we have $\chi\lambda=-\chi$. Therefore $\chi\lambda=\chi$ exactly when $\chi$ vanishes outside $H$. This proves all three equivalences in the <index-two character restriction dichotomy>:
$$
\boxed{(1)\Longleftrightarrow(2)\Longleftrightarrow(3).}
$$
Back to article page
