Subhomomorph-containing subgroup
From Groupprops
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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]
WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with sub-homomorph-containing subgroup
Definition
A subgroup of a group
is termed a subhomomorph-containing subgroup if for any subgroup
and any homomorphism of groups
, we have
.
Relation with other properties
Stronger properties
Weaker properties
- Homomorph-containing subgroup
- Subisomorph-containing subgroup
- Subhomomorph-dominating subgroup
- Transfer-closed fully invariant subgroup
- Intermediately fully invariant subgroup
- Fully invariant subgroup
- Intermediately strictly characteristic subgroup
- Strictly characteristic subgroup
- Transfer-closed characteristic subgroup
- Intermediately characteristic subgroup
- Characteristic subgroup
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
Suppose are groups such that
is a subhomomorph-containing subgroup of
and
is a subhomomorph-containing subgroup of
. Then,
is a subhomomorph-containing subgroup of
. For full proof, refer: Subhomomorph-containment is transitive
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
Suppose are groups such that
is a subhomomorph-containing subgroup of
. Then,
is a subhomomorph-containing subgroup of
. For full proof, refer: Subhomomorph-containment satisfies intermediate subgroup condition