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

This article describes a computation relating the result of the Join operator (?) on two known subgroup properties (i.e., Characteristic subgroup (?) and Characteristic-potentially characteristic subgroup (?)), to another known subgroup property (i.e., Characteristic-potentially characteristic subgroup (?))
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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.

Definitions used

Characteristic-potentially characteristic subgroup

Further information: Characteristic-potentially characteristic subgroup

Related facts

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. Characteristicity is transitive
2. 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 $G$.