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).
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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 .
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 .