Subgroup whose normal closure is homomorph-containing

From Groupprops
Jump to: navigation, search
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]


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

Definition

A subgroup H of a group G is termed a subgroup whose normal closure is homomorph-containing if H^G, its normal closure in the whole group G, is a homomorph-containing subgroup of G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
homomorph-dominating subgroup any subgroup of the whole group that is a homomorphic image of the subgroup is contained in one of its conjugates Join of homomorph-dominating subgroups|FULL LIST, MORE INFO
Sylow subgroup subgroup of finite group with maximal prime power order (via homomorph-dominating) Join of Sylow subgroups|FULL LIST, MORE INFO
join of Sylow subgroups join of Sylow subgroups follows from statement for Sylow subgroups, and homomorph-containment is strongly join-closed Join of homomorph-dominating subgroups|FULL LIST, MORE INFO
Hall subgroup order and index are relatively prime (via join of Sylow subgroups) Join of Sylow subgroups, Join of homomorph-dominating subgroups|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup whose normal closure is fully invariant normal closure is a fully invariant subgroup follows from homomorph-containing implies fully invariant follows from fully invariant not implies homomorph-containing, and the fact that since fully invariant implies normal, a fully invariant subgroup is its own normal closure |FULL LIST, MORE INFO
subgroup whose normal closure is characteristic normal closure is a characteristic subgroup (via subgroup whose normal closure is fully invariant, using fully invariant implies characteristic) (via subgroup whose normal closure is fully invariant) Subgroup whose normal closure is fully invariant|FULL LIST, MORE INFO