I'm stuck on an exercise in Isaacs's book "Character Theory of Finite Groups" - it relates to something I'm looking at as part of ongoing research, but I guess it belongs here rather than on MathOverflow, since it's an exercise and hence ought to be solvable by basic techniques...
Anyway, the question/problem is Exercise 5.14(a), and goes as follows.
Let G be a finite nonabelian group and let $f$ be the smallest character degree that is not 1. Suppose that the derived subgroup (a.k.a. the commutator subgroup) has order $\leq f$. Prove that the derived subgroup is contained in the centre of $G$.
The hint given is the (fairly obvious) fact that each conjugacy class injects into $G'$, so by the assumption of the problem is bounded above by $f$. This makes me think that the solution has something to do with column orthogonality in the character table, using the fact that the linear characters all take the value 1 on the derived subgroup; but I haven't managed to make that idea work.
If it helps or makes any difference: the chapter for which this is an exercise is the one introducing induced characters (but precedes the discussion of induction from normal subgroups, and Clifford theory). Again, I can't see a way to induce anything from the derived subgroup to make progress: my only idea would be to take a non-trivial irreducible character on the derived subgroup, induce it to a character on $G$, and observe that the induced character is orthogonal to all the linear characters of $G$ by e.g. Frobenius reciprocity.
I'm sure I'm just missing something obvious, so would be happy with just a hint or two.