Amalgam-characteristic subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Amalgam-characteristic subgroup, all facts related to Amalgam-characteristic subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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		Amalgam-normal-subhomomorph-containing subgroup, Amalgam-strictly characteristic subgroup, Periodic normal subgroup|FULL LIST, MORE INFO
periodic normal subgroup	normal subgroup and every element has finite order	periodic normal implies amalgam-characteristic		Amalgam-normal-subhomomorph-containing subgroup, Amalgam-strictly characteristic subgroup|FULL LIST, MORE INFO
central subgroup	contained in the center, i.e., its elements commute with every element	central implies amalgam-characteristic		Amalgam-strictly characteristic subgroup, Normal subgroup contained in the hypercenter|FULL LIST, MORE INFO
normal subgroup contained in the hypercenter	normal and contained in the hypercenter	Normal subgroup contained in hypercenter is amalgam-characteristic		Amalgam-strictly characteristic subgroup|FULL LIST, MORE INFO
amalgam-normal-subhomomorph-containing subgroup	normal-subhomomorph-containing subgroup in the amalgam of the whole group over it			Amalgam-strictly characteristic subgroup|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).