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}



Beyond Brauer's theorem
Jeremy Rickard's answer is perfectly correct, of course. It is also easy to see (by an argument close to Taketa's) that if $G$ is a finite simple group, and $\chi$ is a faithful (not assumed irreducible, though in fact it will be irreducible) complex character of minimal degree of $G$, then $\chi$ is not a non-negative integer combination of characters induced from proper subgroups of $G$. For if $\chi = \sum\_{\mu} {\rm Ind}\_{H(\mu)}^{G}(\mu),$ where $\mu$ is a character of the proper subgroup $H(\mu)$ of $G$, then there must be a character $\mu$ such that ${\rm Ind}\_{H(\mu)}^{G}(\mu)$ is not a multiple of the trivial character. Then (using Frobenius reciprocity)
$\theta = {\rm Ind}\_{H(\mu)}^{G}(1) -1$ is a character whose kernel is a normal subgroup of $G$ contained in $H(\mu)$, so $\theta$ is faithful. But certainly $\theta$ has smaller degree than $\chi$, contrary to the minimality of $\chi(1)$.


While they do not address non-negativity issues, there are interesting books on so-called Explicit Brauer Induction ( separate books by R. Boltje and V. Snaith taking a "dual" viewpoints) which discuss minimal families which can be used when seeking induction theorems which commute with restriction to subgroups (in Boltje's case) or induction from subgroups (in Snaith's case).


In a related direction, I remember being present at a discussion where Walter Feit stated that it is the case that whenever $\chi$ is an irreducible complex character of a finite group $G$, there is a finite overgroup $H$ of $G$ and a monomial irreducible character $\theta$ of $H$ such that $\chi$ occurs as a constituent of ${\rm Res}^{H}\_{G}(\theta)$. I have no idea how such a statement would be proved, and it may require the classification of finite simple groups. It is related to generalizing in some way to arbitrary finite groups a Theorem of E.C. Dade that every finite solvable group $G$ can be embedded in an $M$-group ( an $M$-group is a finite group all of whose irreducible characters are induced from linear characters of (not necessarily proper) subgroups).


