Amalgam-characteristic subgroup

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Definition

Definition with symbols

Suppose $G$ is a group and $H$ is a subgroup of $G$. We say that $H$ is an amalgam-characteristic subgroup of $G$ if $H$ is characteristic in the group $L$ given by:

$L=G*_{H}G$.

In other words, $L$ is the amalgam of $G$ with itself over $H$.

Relation with other properties

Stronger properties

• Finite normal subgroup: For full proof, refer: Finite normal implies amalgam-characteristic
• Central subgroup: For full proof, refer: Central implies amalgam-characteristic
• Normal subgroup contained in a member of the upper central series: For full proof, refer: Normal subgroup in upper central series member is amalgam-characteristic

Weaker properties

• Potentially characteristic subgroup: For full proof, refer: Amalgam-characteristic implies potentially characteristic
• Normal subgroup: For proof of the implication, refer Amalgam-characteristic implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies amalgam-characteristic.