= Solution
For any $h\in G$, left multiplication permutes the elements of the <finite group>. Consequently <averaging over a finite group> gives
$$
R(h)Pv=\frac1{|G|}\sum_{g\in G}R(hg)v=\frac1{|G|}\sum_{k\in G}R(k)v=Pv.
$$
Thus $Pv$ belongs to the <fixed-point subspace of a group action> $V^G$; any nonzero such vector spans a <trivial representation>, or singlet. The zero vector is included in the invariance statement.
Conversely $Pw=w$ for every $w\in V^G$, so
$$
\boxed{\operatorname{im}P=V^G,\qquad P^2=P.}
$$
This is a <projection operator> onto all singlets, rather than necessarily onto a one-dimensional <vector subspace>. Its <trace> is the singlet multiplicity $|G|^{-1}\sum_g\chi_R(g)$.
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