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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Definition

A subgroup $H$ of a group $G$ is termed homomorph-containing if for any $\varphi \in \operatorname {Hom} (H,G)$, the image $\varphi (H)$ is contained in $H$.

Relation with other properties

Stronger properties

• Order-containing subgroup
• Subhomomorph-containing subgroup
• Variety-containing subgroup
• Normal Sylow subgroup
• Normal Hall subgroup
• Fully invariant direct factor
• Left-transitively homomorph-containing subgroup
• Right-transitively homomorph-containing subgroup

Weaker properties

• Fully invariant subgroup: Also related:
  • Intermediately fully invariant subgroup
  • Strictly characteristic subgroup
  • Intermediately characteristic subgroup
  • Characteristic subgroup
  • Normal subgroup
• Isomorph-containing subgroup
• Homomorph-dominating subgroup

Facts

• The omega subgroups of a group of prime power order are homomorph-containing. Further information: Omega subgroups are homomorph-containing

Metaproperties

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).
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For any group $G$, the trivial subgroup and the whole group are both homomorph-containing.

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

We can have subgroups $H\leq K\leq G$ such that $H$ is a homomorph-containing subgroup of $K$ and $K$ is a homomorph-containing subgroup of $G$ but $H$ is not a homomorph-containing subgroup of $G$. For full proof, refer: Homomorph-containment is not 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.
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If $H\leq K\leq G$ and $H$ is a homomorph-containing subgroup of $G$, $H$ is also a homomorph-containing subgroup of $K$. For full proof, refer: Homomorph-containment satisfies intermediate subgroup condition

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

If $H_{i},i\in I$, are all homomorph-containing subgroups of $G$, then so is the join of subgroups $\langle H_{i}\rangle _{i\in I}$. For full proof, refer: Homomorph-containment is strongly join-closed

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
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If $H\leq K\leq G$ are groups such that $H$ is a homomorph-containing subgroup of $G$ and $K/H$ is a homomorph-containing subgroup of $G/H$, then $K$ is a homomorph-containing subgroup of $G$. For full proof, refer: Homomorph-containment is quotient-transitive