Let $G$ be a finite group. The regular $\chi_{reg}$ of $G$ is the character corresponding to representation of $G$ on $\mathbb{C}[G]$ with action by (left) multiplication to basis elements of $G$.
It is well known that $\chi_{reg}$ is zero at all $g\in G$ except $g=1$. Also the degree of this character (i.e. value at $1$) is $|G|$.
There is an exercise in Isaacs' character theory about similar kind of character:
Fact: Let $G$ be a finite group and $\chi$ a character of $G$ such that $\chi$ vanishes on $G-\{1\}$. Then $|G|$ divides degree of character.
Question: In the above fact, can we also say that $\chi$ contains $\chi_{reg}$ as a component? (i.e., is $\chi$ equal to sum of $\chi_{reg}$ and some other character of $G$?
Note: In above discussion, character of $G$ means character of a complex (possibly reducible) representation of $G$, and not virtual character, or class function etc.
This character $\chi$ is a multiple of the character of the regular representation: $\chi=a\chi_{reg}$. A priori $a\in\Bbb Q$, but the regular representation contains one copy of the trivial representation. So $\chi$ contains $a$ copies of the trivial representation, so $a\in\Bbb Z$. Therefore the degree of $\chi$ is $a|G|$.