Conjugacy-closedness is not upper join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugacy-closed subgroup) not satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about conjugacy-closed subgroup|Get more facts about upper join-closed subgroup property|
Statement
We can have the following situation: is a subgroup,
are intermediate subgroups of
containing
, such that
is conjugacy-closed in
as well as in
, but not in the join
.
Proof
A generic example
Suppose is a group with an automorphism
of order two that is not class-preserving: the automorphism does not preserve conjugacy classes. For instance,
could be an Abelian group of exponent greater than two, and
could be the inverse map.
Let be the subgroup of
generated by the automorphism
(i.e.,
acting coordinate-wise) and the coordinate exchange automorphism. Since both these automorphisms are of order two and commute,
is a Klein-four group. Define
with the specified action.
Now, let be the two-element subgroup of
generated by the coordinate exchange automorphism, and
be the two-element subgroup of
generated by the composite the coordinate exchange automorphism and
. Define
and
.
- The subgroup
is conjugacy-closed in
: Any element of
is a product of an element in
and an element in
. Note that the element of
preserves conjugacy classes, so it remains to see the effect of the element of
. if the element of
is trivial, it of course acts as the identity. If it is not trivial, it sends every element of
to an element of
, so no two distinct conjugacy classes in
get fused by the action of
.
- The subgroup
is conjugacy-closed in
: The reasoning is identical to the above reasoning.
- The subgroup
is not conjugacy-closed in
: Indeed, the automorphism
does not preserve conjugacy classes in
.
A similar kind of example can be constructed when the automorphism is not class-preserving, and has finite order
. In this case, we need to case a
-fold direct product of
, and have the diagonal automorphism
as well as the cyclic coordinate permutation automorphism act on this direct product.
Some particular examples
The smallest particular example of the above is when is the cyclic group of order three, and
is the inverse map.
in this case has order
, and
both have order
.
Within nilpotent groups, the smallest particular example is when is the cyclic group of order four, and
is the inverse map.
has order
, and
both have order
.