Central implies potentially fully invariant in finite

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Central subgroup (?)) must also satisfy the second subgroup property (i.e., Potentially fully invariant subgroup (?)). In other words, every central subgroup of finite group is a potentially fully invariant subgroup of finite group.
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Central subgroup (?)) must also satisfy the second subgroup property (i.e., Finite-potentially fully invariant subgroup (?)). In other words, every central subgroup of finite group is a finite-potentially fully invariant subgroup of finite group.
View all subgroup property implications in finite groups $|$ View all subgroup property non-implications in finite groups $|$ View all subgroup property implications $|$ View all subgroup property non-implications

Statement

Suppose $G$ is a finite group and $H$ is a central subgroup of $G$. In other words, $H$ is a subgroup of $G$ contained in the center of $G$. Then, there exists a group $K$ containing $G$ such that $H$ is a fully invariant subgroup of $K$.

Related facts

Stronger facts

• Central implies finite-pi-potentially verbal in finite
• Central implies potentially verbal in finite
• Central implies finite-pi-potentially fully invariant in finite

Other related facts

• Central implies finite-pi-potentially characteristic in finite
• Cyclic normal implies finite-pi-potentially verbal in finite, which implies that cyclic normal implies potentially fully invariant in finite
• Homocyclic normal implies finite-pi-potentially fully invariant in finite, which implies that homocyclic normal implies potentially fully invariant in finite