Let
G
G
be a reductive linear algebraic group over an algebraically closed field of characteristic
p
>
0
p > 0
. A subgroup of
G
G
is said to be separable in
G
G
if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre’s concept of
G
G
-complete reducibility for subgroups of
G
G
. A separability hypothesis appears in many general theorems concerning
G
G
-complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if
G
G
is a connected reductive group and
p
p
is very good for
G
G
, then any subgroup of
G
G
is separable; we deduce that under these hypotheses on
G
G
, a subgroup
H
H
of
G
G
is
G
G
-completely reducible provided Lie
G
G
is semisimple as an
H
H
-module.
Recently, Guralnick has proved that if
H
H
is a reductive subgroup of
G
G
and
C
C
is a conjugacy class of
G
G
, then
C
∩
H
C\cap H
is a finite union of
H
H
-conjugacy classes. For generic
p
p
— when certain extra hypotheses hold, including separability — this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig’s deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick’s result is false if one considers conjugacy classes of
n
n
-tuples of elements from
H
H
for
n
>
1
n > 1
.