Let be a group. A subgroup
of
is called characteristic if
for every automorphism of
of
Compare that to the definition of a normal subgroup: a subgroup
of
is normal if
for every inner automorphism
of
So every characteristic subgroup is normal. A few examples of characteristic subgroups of
are: the trivial subgroup
the group
itself, the center of
and the commutator subgroup of
Remark. Let be a subgroup of a group
If
is an automorphism of
then
is also an automorphism of
So
is characteristic if and only if
for all automorphisms
of
Definition 1. A group is called characteristically simple if the only characteristic subgroups of
are
and
itself.
In this post, we characterize all finite characteristically simple groups. But first, let’s give a few examples of those groups. The trivial ones are simple groups; it is clear that every simple group is characteristically simple because every characteristic subgroup is normal. A highly non-trivial example is given in Example 2 in this post. A slightly non-trivial examples of characteristically simple groups are finite elementary abelian -groups. Recall that a finite elementary abelian
–group, where
is a prime number, is any finite direct product of copies of the cyclic group
Looking at the group additively, it is clear that finite elementary abelian
-groups are just finite dimensional vector spaces over the finite field
(or you can call this one
if you have trouble distinguishing between
as a cyclic group and
as a field).
Example 1. Every finite elementary abelian -group
is characteristically simple.
Proof. So, considering additively,
is a finite dimensional vector space over the field
Suppose, to the contrary, that
has a characteristic subgroup
Let
be a basis for
and extend it to a basis
of
Now, the
-linear map
defined by
is an automorphism of and so, since
is characteristic,
contradiction.
Definition 2. A minimal normal subgroup of a group is a normal subgroup
of
such that the only subgroups of
that are normal in
are
and
itself.
Example 2. Every characteristic subgroup of a normal subgroup is normal. In particular, every minimal normal subgroup of a group is characteristically simple.
Proof. Let be a characteristic subgroup of a normal subgroup
of a group
and let
Then, since
is a normal subgroup of
the map
defined by
is an automorphism of
Since
is a characteristic subgroup of
we have
and so
for all
i.e.
is normal in
We are now ready to characterize all finite characteristically simple groups.
Theorem. A finite group is characteristically simple if and only if
is the direct product of some isomorphic simple groups.
Proof. Suppose first that
where is a simple group. We need to show that
is characteristically simple. Since
is simple, either
is abelian or
the center of
is trivial. If
is abelian, then
for some prime
and hence
is characteristically simple by Example 1. So we may assume that
Suppose that
is a characteristic subgroup of
and let
Then
and so there exists
such that
Let
Then, since
is normal,
Notice that since we have
So we have shown the set
which is a normal subgroup of is non-trivial and hence
i.e.,
for all
So
for all
and all
because the map
defined by
is an automorphism of and
is characteristic. Thus
and the proof of one side of the Theorem is complete. Notice that for this part of the Theorem, we did not need
to be finite.
Conversely, suppose that is characteristically simple. We need to show that
is the direct product of some isomorphic simple groups. Choose a normal subgroup
of
such that
and
is as small as possible. Let
be subgroups of
such that
and the positive integer
is as large as possible.
Claim 1. Let be an automorphism of
Then
for all
Proof. First notice that since for all
each
hence
is a normal subgroup of
and
for all
Suppose now, to the contrary, that
for some
So
is a proper subgroup of
and thus
Hence, since
is a normal subgroup of
we have
by minimality of
So
are two normal subgroups of
whose intersection is trivial. That implies
which contradicts maximality of
This contradiction completes the proof of Claim 1.
Claim 2.
Proof. By Claim 1, for all
and so
for all automorphisms
of
Therefore
is a characteristic subgroup of
and hence
because
and
is characteristically simple. The proof of Claim 2 is now complete.
We can now finish the proof of the Theorem. By Claim 2, for some normal subgroups
such that
for all
So the only thing left is to show that each
is simple. Suppose that
is a normal subgroup of some
Then
is also a normal subgroup of
because
is a direct summand of
But
and so, by minimality of
we must have
or
Hence either
or
which proves that
is simple.
Corollary. Let be a minimal normal subgroup of a finite group
Then either
is an elementary abelian
-group or
is the direct product of some isomorphic non-abelian simple groups.
Proof. By Example 2, is characteristically simple and hence, by the Theorem,
is the direct product of some isomorphic simple groups
We are done if each
is non-abelian. Suppose now that
is abelian for some
Then
for some prime
because
is simple and abelian. Hence
for all
and thus
is an elementary abelian
-group. in this case.
Note. The reference for the only if part of the Theorem and the Corollary is chapter 2, section 1, of the book Finite Groups by Daniel Gorenstein. Strangely, Gorenstein doesn’t even mention the if part of the Theorem in his book; the proof of that was added by me.
Exercise 1. Show that every characteristically simple group of order
is simple.
Hint. If is not simple, then, by the Theorem,
for some simple group
and some integer
Show that the equation
has no solution for integers
by, for example, proving that
has alway a prime divisor of exponent
Exercise 2. Show that a finite solvable group is characteristically simple if and only if it is an elementary abelian
-group.
Hint. Since is solvable,
and so if
is characteristically simple, then
i.e.
is abelian. Now use the Theorem.
Exercise 3. Show that the symmetric group is not characteristically simple.
Hint. First Proof: The alternating group is a normal subgroup of
and so
is not simple hence not characteristically simple, by Exercise 1. Second Proof:
is the unique subgroup of order
in
and hence
for any automorphism
of
i.e.
is a characteristic subgroup of