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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 with symbols
In other words, is the amalgam of with itself over , and is treated as the subgroup of given by the amalgamated between the two factors.
Relation with other 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
- 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.
- 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).