Faithful character of degree less than p gives abelian p-Sylow groups.

Question

I am trying to prove:

Suppose $p$ divides $|G|$ and let $\chi \in Irr(G)$ if $\chi$ is faithful and its degree is less than $p$ then any $p$-Sylow subgroup of $G$ is abelian.

I have tried to decompose the restriction of $\chi$ to $P$, a $p$-group, into irreducibles and I guess I should try to prove that this gives me all one dimensional constituents, but I have no idea how to proceed.

Any hints, links to good sources or other help is much appreciated.

Thanks.

Answer 1

OK, I'll make it an answer. The restriction $\chi_P$ of $\chi$ to $P \in {\rm Sy}_p(G)$ decomposes into irreducible characters of $P$. Since the degree of any such character divides $|P|$, it must be power of $p$. But $\deg(\chi) < p$, so all of the irreducible constituents of $\chi_P$ have degree $1$. Since $\chi$ and hence $\chi_P$ is faithful, so $P$ must be abelian.

By the way, there is no need to assume that $\chi$ is irreducible, just that it is faithful.