Sub-homomorph-containing 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, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with subhomomorph-containing subgroup

Definition

A subgroup $H$ of a group $G$ is termed a sub-homomorph-containing subgroup if there is a chain of subgroups:

$H = H_1 \le H_2 \le \dots \le H_n = G$

such that each $H_i$ is a homomorph-containing subgroup of $H_{i+1}$.

Formalisms

In terms of the subordination operator

This property is obtained by applying the subordination operator to the property: homomorph-containing subgroup
View other properties obtained by applying the subordination operator

Relation with other properties

Stronger properties

• Homomorph-containing subgroup
• Subhomomorph-containing subgroup
• Normal Sylow subgroup
• Normal Hall subgroup

Weaker properties

• Fully invariant subgroup
• Strictly characteristic subgroup
• Sub-isomorph-containing subgroup
• Characteristic subgroup
• Normal 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

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties