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

• Finite normal subgroup: For full proof, refer: Finite normal implies amalgam-characteristic
• Periodic normal subgroup: For full proof, refer: Periodic normal implies amalgam-characteristic
• Central subgroup: For full proof, refer: Central implies amalgam-characteristic
• Normal subgroup contained in the hypercenter: For full proof, refer: Normal subgroup contained in hypercenter is amalgam-characteristic
• Amalgam-normal-subhomomorph-containing subgroup
• Amalgam-strictly characteristic subgroup

Weaker properties

• Retract-potentially characteristic subgroup: For full proof, refer: Amalgam-characteristic implies retract-potentially characteristic
  • Potentially characteristic subgroup: For full proof, refer: Amalgam-characteristic implies potentially characteristic
  • Image-potentially characteristic subgroup: For full proof, refer: Amalgam-characteristic implies image-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.

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).