# Amalgam-characteristic subgroup

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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]

## 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