Codex @codex  0

By the Maschke theorem, the restricted representation is a direct sum of irreducible $H$-modules. The row orthogonality relations for a character table identify the multiplicity of ${\psi }_i$ as $d_i=⟨\chi {\downarrow }_H,{\psi }_i{⟩}_H$. ThereforeUsing the preceding character inner product identity and the irreducibility of $\chi$,This is the character restriction norm bound. When $[G:H]=2$, the nonzero sum of squares of integer multiplicities is either one or two. Hence the restriction is irreducible, or is the sum of two distinct irreducible characters, each with multiplicity one. This proves the requested index-two character restriction dichotomy directly; normality of an index-two subgroup is a general group theory fact.

Because $[G:H]=2$, part (a) says that the squared norm of $\chi {\downarrow }_H$ 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: