Normal-potentially 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.
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This term is related to: potentially characteristic subgroups characterization problem
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Definition
A subgroup H of a group K is termed normal-potentially characteristic in K if there exists a group G containing K such that:
- K is a normal subgroup of G.
- H is a characteristic subgroup of G.
Formalisms
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
In terms of the upper-hook operator
Given two subgroup properties p and q, the upper-hook operator of p and q is defined as the following property r: a subgroup H of a group K has property r if there exists a group G containing K such that H has property p in G and K has property q in G.
The property of being semi-strongly potentially characteristic is thus obtained by applying the upper-hook operator to the properties characteristic subgroup and normal subgroup.
Relation with other properties
Stronger properties
Weaker properties
- Normal-potentially relatively characteristic subgroup
- Normal-extensible automorphism-invariant subgroup
- Normal subgroup
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