Join of characteristic and characteristic-potentially characteristic implies characteristic-potentially characteristic

Statement
Suppose $$H,K$$ are subgroups of a group $$G$$ such that $$H$$ is a characteristic subgroup of $$G$$ and $$K$$ is a characteristic-potentially characteristic subgroup of $$G$$. Then, the join of subgroups $$\langle H, K \rangle$$ (which in this case is equal to the product of subgroups $$HK$$) is also a characteristic-potentially characteristic subgroup.

Similar facts
About joins:


 * Join of characteristic and normal-potentially relatively characteristic implies normal-potentially relatively characteristic

About intersections:


 * Intersection of characteristic and characteristic-potentially characteristic implies characteristic-potentially characteristic
 * Intersection of characteristic and normal-potentially relatively characteristic implies normal-potentially relatively characteristic

About composition (subgroups of subgroups):


 * Characteristic of potentially characteristic implies potentially characteristic
 * Characteristic of characteristic-potentially characteristic implies characteristic-potentially characteristic
 * Characteristic of normal-potentially characteristic implies normal-potentially characteristic
 * Characteristic of normal-potentially relatively characteristic implies normal-potentially relatively characteristic

Facts used

 * 1) uses::Characteristicity is transitive
 * 2) uses::Characteristicity is strongly join-closed

Proof
Given: A group $$G$$, subgroups $$H,K$$ of $$G$$ such that $$H$$ is characteristic in $$G$$ and $$K$$ is characteristic-potentially characteristic in $$G$$.

To prove: The join $$HK$$ is also characteristic-potentially characteristic in $$G$$.

Proof: By the definition of characteristic-potentially characteristic, there is a group $$L$$ containing $$G$$ such that both $$K$$ and $$G$$ are characteristic in $$L$$.


 * 1) $$H$$ is characteristic in $$L$$: $$H$$ is characteristic in $$G$$ and $$G$$ is characteristic in $$L$$, so fact (1) yields that $$H$$ is characteristic in $$L$$.
 * 2) $$HK$$ is characteristic in $$L$$: $$H$$ and $$K$$ are both characteristic in $$L$$, so by fact (2), so is $$HK$$.

Thus, $$L$$ is a group containing $$G$$ such that both $$HK$$ and $$G$$ are characteristic in $$L$$. Thus, $$HK$$ is characteristic-potentially characteristic in $$G$$.