Characteristic subgroup
A characteristic subgroup of a group is a subgroup of
that is stable under every automorphism on
. Since the map
is an automorphism (specifically, an inner automorphism) on
, for every
, it follows that every characteristic subgroup of
is also a normal subgroup of
.
Examples
Every group is a characteristic subgroup of itself; a group's trivial subgroup is characteristic.
Let be a natural number that divides the order of
. Then the set of elements
of
for which
divides
is a characteristic subgroup of
. Since every subgroup of a cyclic group is cyclic, it follows that every subgroup of a cyclic group is a characteristic group.
In general, though, not every cyclic group of an Abelian group is characteristic. For instance, the Klein 4-group has no non-trivial characteristic subgroups, since any permutation of its non-identity elements is an automorphism. For an odd prime , the group
has no nontrivial characteristic subgroups either. Indeed, for any
relatively prime to
, the mapping
is an automorphism, as is the mapping
. Thus if
(
) is a member of a characteristic subgroup, then so is
, and these two evidently generate
; and if
(
) is an element of a characteristic subroup, then setting
, we see that
is an element of this characteristic subgroup; therefore so is all of
.
This idea also shows that has no non-trivial characteristic subgroups, for any natural number
. In fact, the characteristic subgroups of
are of the form
, for some integer
.
Characteristic Subgroups of Normal Subgroups
Theorem 1. Let be a group, and let
be a normal subgroup of
. Let
be a characteristic subgroup of
. Then
is a normal subgroup of
; furthermore, if
is a characteristic subgroup of
, then so is
.
Proof. Let be an (inner) automorphism on
. Then its restriction to
is an automorphism on
, so
is characteristic (normal) in
if and only if
is.
Theorem 2. An equivalence relation is compatible with the group law on
and every automorphism on
if and only if
is equivalent to
, for some characteristic subgroup
of
.
Proof. Evidently, must be of the form
, where
is the set of elements equivalent to
under
. For any
and any automorphism
on
,
, so
is a characteristic subgroup.
On the other hand, if is a characteristic subgroup, the relation
is compatible with the group law; it also implies
.