  In 1878, Jordan proved that if a finite group $G$ has a faithful
representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal
abelian subgroup with index bounded above by a function of $n$. The same result
fails if one replaces $\mathbb{C}$ by a field of positive characteristic, due
to the presence of large unipotent and/or Lie type subgroups. For this reason,
a long-standing problem in group and representation theory has been to find the
"correct analogue" of Jordan's theorem in characteristic $p>0$. Progress has
been made in a number of different directions, most notably by Brauer and Feit
in 1966; by Collins in 2008; and by Larsen and Pink in 2011. With a 1968
theorem of Steinberg in mind (which shows that a significant proportion of
elements in a simple group of Lie type are unipotent), we prove in this paper
that if a finite group $G$ has a faithful representation over a field of
characteristic $p$, then a significant proportion of the elements of $G$ must
have $p$-power order. We prove similar results for permutation groups, and
present a general method for counting $p$-elements in finite groups. All of our
results are best possible.
