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Finite nonabelian groups with few positive real character values?

This question was just raised by a colleague (who shall for the moment remain anonymous). It may or may not have a reasonable answer.

For which finite nonabelian groups $G$ do all irreducible complex characters of degree $>1$ have at most one strictly positive real value (namely the degree)?

One small example is the alternating group $A_4$. This could be viewed as a sort of degenerate Chevalley group, whose Steinberg character of degree $3$ has other values $-1, 0, 0$. But for most finite groups of Lie type over a finite field of order $q$, the Steinberg character of degree a power of $q$ will take on more than one positive integral value. While an enumeration of finite groups having the stated property may well be out of reach, it would be of interest to know:

Are there infinitely many groups $G$ with the special property stated above?
show/hide this revision's text 1

Finite groups with few positive real character values?

This question was just raised by a colleague (who shall for the moment remain anonymous). It may or may not have a reasonable answer.

For which finite groups $G$ do all irreducible complex characters of degree $>1$ have at most one strictly positive real value (namely the degree)?

One small example is the alternating group $A_4$. This could be viewed as a sort of degenerate Chevalley group, whose Steinberg character of degree $3$ has other values $-1, 0, 0$. But for most finite groups of Lie type over a finite field of order $q$, the Steinberg character of degree a power of $q$ will take on more than one positive integral value. While an enumeration of finite groups having the stated property may well be out of reach, it would be of interest to know:

Are there infinitely many groups $G$ with the special property stated above?