= Generalized character separating complementary prime elements
{title2=$\xi(a)=1,\quad\xi(b)=0$}
Suppose every nonidentity element of a finite group $K$ is either a <pi-element> or an element for complementary primes. An <elementary subgroup> cannot have nontrivial factors of both kinds, since the product of nonidentity commuting coprime-order elements would have mixed prime order. Choose an integer $d$ congruent to $1$ modulo $|K|_\pi$ and to $0$ modulo $|K|_{\pi'}$. Set $f(1)=d$, $f=1$ on nonidentity pi-elements and $f=0$ on the other nonidentity elements. On each elementary subgroup this is an integer combination of the trivial and regular characters. <Brauer's characterization of characters> supplies the asserted <generalized character>.
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