Characteristicity is not upper join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).
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Contents
Statement
Property-theoretic statement
The subgroup property of being a characteristic subgroup does not satisfy the subgroup metaproperty of being upper join-closed.
Statement with symbols
It is possible to find a group , a subgroup
of
, and intermediate subgroups
of
containing
, such that
is characteristic in both
and
, but
is not characteristic in
.
Proof
A generic example
Let be a group and
be subgroups of
such that:
-
- There is no nontrivial homomorphism from
to
- There is no nontrivial homomorphism from
to
Then, set . We then observe that:
-
is a characteristic subgroup inside
(in fact,
is fully characteristic inside
).
-
is a characteristic subgroup inside
(in fact,
is fully characteristic inside
).
-
is not characteristic inside
.
Note that this generic example also shows the the property of being a fully characteristic subgroup is not closed under upper joins. For full proof, refer: Full characteristicity is not upper join-closed
Some specific examples
One example is to take as a simple group, and
as proper subgroups generating it. For instance,
is the alternating group on five letters, and
can be taken, for instance, as two of the embedded alternating groups on four letters.
Another generic example
Let be a non-Abelian group generated by Abelian subgroups
. Consider:
-
-
-
-
.
Then, is the commutator subgroup both of
and of
, so
is characteristic in both. On the other hand,
is not characteristic in
because the coordinate exchange automorphism (that swaps the two copies of
) does not preserve
.
This generic example also shows that the property of being fully characteristic is not preserved under upper joins. For full proof, refer: Full characteristicity is not upper join-closed
An example is a non-Abelian group of order
, and
and
two Abelian subgroups that generate it.