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 $H$, and $H$ is treated as the subgroup of $L$ given by the amalgamated $H$ between the two factors.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite normal subgroup	normal subgroup that is finite as a group	finite normal implies amalgam-characteristic		|FULL LIST, MORE INFO
periodic normal subgroup	normal subgroup and every element has finite order	periodic normal implies amalgam-characteristic		|FULL LIST, MORE INFO
central subgroup	contained in the center, i.e., its elements commute with every element	central implies amalgam-characteristic		|FULL LIST, MORE INFO
normal subgroup contained in the hypercenter	normal and contained in the hypercenter	Normal subgroup contained in hypercenter is amalgam-characteristic		|FULL LIST, MORE INFO
amalgam-normal-subhomomorph-containing subgroup	normal-subhomomorph-containing subgroup in the amalgam of the whole group over it			|FULL LIST, MORE INFO
amalgam-strictly characteristic subgroup	strictly characteristic subgroup in the amalgam of the whole group over it			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup		amalgam-characteristic implies normal	normal not implies amalgam-characteristic	|FULL LIST, MORE INFO

Incomparable properties

• Characteristic subgroup: For full proof, refer: Characteristic not implies amalgam-characteristic Note that the other non-implication is clear from, for instance, finite normal implies amalgam-characteristic, because not every finite normal subgroup is characteristic (For full proof, refer: Normal not implies characteristic).