Conjugacy-closedness is not upper join-closed

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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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Statement

We can have the following situation: H \le G is a subgroup, K_1, K_2 are intermediate subgroups of G containing H, such that H is conjugacy-closed in K_1 as well as in K_2, but not in the join \langle K_1, K_2.

Proof

A generic example

Suppose H is a group with an automorphism \sigma of order two that is not class-preserving: the automorphism does not preserve conjugacy classes. For instance, H could be an Abelian group of exponent greater than two, and \sigma could be the inverse map.

Let A be the subgroup of \operatorname{Aut}(H \times H) generated by the automorphism (\sigma,\sigma) (i.e., \sigma acting coordinate-wise) and the coordinate exchange automorphism. Since both these automorphisms are of order two and commute, A is a Klein-four group. Define G = (H \times H) \rtimes A with the specified action.

Now, let B_1 be the two-element subgroup of A generated by the coordinate exchange automorphism, and B_2 be the two-element subgroup of A generated by the composite the coordinate exchange automorphism and (\sigma,\sigma). Define K_1 = (H \times H)B_1 and K_2 = (H \times H)B_2.

  • The subgroup H = H \times \{ e \} is conjugacy-closed in K_1: Any element of K_1 is a product of an element in H \times H and an element in H \times H. Note that the element of H \times H preserves conjugacy classes, so it remains to see the effect of the element of B_1. if the element of B_1 is trivial, it of course acts as the identity. If it is not trivial, it sends every element of H \times \{ e \} to an element of \{ e \} \times H, so no two distinct conjugacy classes in H \times \{ e \} get fused by the action of B_1.
  • The subgroup H = H \times \{ e \} is conjugacy-closed in K_1: The reasoning is identical to the above reasoning.
  • The subgroup H = H \times \{ e \} is not conjugacy-closed in G: Indeed, the automorphism (\sigma,\sigma) does not preserve conjugacy classes in H \times \{ e \}.

A similar kind of example can be constructed when the automorphism \sigma is not class-preserving, and has finite order n > 2. In this case, we need to case a n-fold direct product of H, and have the diagonal automorphism \sigma 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 H is the cyclic group of order three, and \sigma is the inverse map. G in this case has order 36, and K_1, K_2 both have order 18.

Within nilpotent groups, the smallest particular example is when H is the cyclic group of order four, and \sigma is the inverse map. G has order 64, and K_1, K_2 both have order 32.