Abelian-potentially characteristic subgroup
From Groupprops
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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
A subgroup of an abelian group
is termed an abelian-potentially characteristic subgroup if there exists an abelian group
containing
such that
is a characteristic subgroup of
.
Relation with other properties
Stronger properties
property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|
Characteristic subgroup of abelian group | characteristic subgroup of abelian group | direct from definition | abelian-potentially characteristic not implies characteristic | |FULL LIST, MORE INFO |
Abelian-potentially fully invariant subgroup | is a fully invariant subgroup in some bigger abelian group | follows from fully invariant implies characteristic | ? | |FULL LIST, MORE INFO |
Abelian-potentially verbal subgroup | is a verbal subgroup in some bigger abelian group | |FULL LIST, MORE INFO | ||
Subgroup of finite abelian group | finite abelian NPC theorem | Abelian-potentially verbal subgroup|FULL LIST, MORE INFO |
Weaker properties
property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|
Abelian-extensible automorphism-invariant subgroup | invariant under all abelian-extensible automorphisms | abelian-potentially characteristic implies abelian-extensible automorphism-invariant | ? | |FULL LIST, MORE INFO |
Subgroup of abelian group | subgroup of an abelian group | Abelian-extensible automorphism-invariant subgroup|FULL LIST, MORE INFO |