Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary subgroups.

This is an improvement over Artin's theorem, in which the linear combination is with *rational* coefficients of characters induced by characters of *cyclic* subgroups.

So we get integer coefficients at the cost of using more complicated subgroups.

It is reasonable to try to improve Brauer's theorem, taking the coeffients to be positive integers. But this doesn't work: there are characters that can't be expressed as linear combination with positive coefficients of characters induced from elementary subgroups, even if we let the coefficients be real numbers (this is exercise 10.5 in Serre's book on representation theory).

In fact, Brauer's result is the best possible in a very concrete sense:

The only groups that have the property that all of its characters are expressible with non-negative coefficients in Brauer's theorem are solvable groups.

p-elementary subgroups is the minimal family of subgroups for which Brauer's theorem holds (Green).

What I was wondering is, if we replace p-elementary subgroups by a bigger (and more complicated) family of subgroups, much like Brauer had to let go of cyclic groups, can we get a similar theorem with positive coefficients that works for all finite groups?

$\require{AMScd}$ \begin{CD} \mathrm{cyclic} @>>> \mathrm{elementary} @>>> \mathrm{?} \\ @V V V\ @VV V\ @VV V\\ \mathbb{Q} @. \mathbb{Z} @. \mathbb{N} \end{CD}

linearcharacters (of elementary subgroups). Taketa's theorem is about non-negative integer linear combinations of characters induced from linear characters of arbitrary subgroups. Clearly the trivial character (or any linear character) of $G$ is not a non-negative integer linear combination of characters induced from proper subgroups. $\endgroup$