= Solution
There are \b[the $p-1$ inflated quotient <characters> of dimension one and the single <representation> $\rho$ of dimension $p-1$] from part (ii). These are distinct: for $p>2$ their dimensions distinguish $\rho$, and for $p=2$ its nontrivial value on $U$ distinguishes it from the trivial quotient <character>. Their dimension squares sum to
$$
(p-1)+(p-1)^2=p(p-1)=|G|.
$$
The sum-of-squares theorem for <irreducible representations> leaves no room for any further irreducible. This includes $p=2$, when $G\cong C_2$ and both <representations> have dimension one.
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