Quotient-subisomorph-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

The following are some equivalent definitions of quotient-subisomorph-containing subgroup.

No.	Shorthand	A subgroup $H$ of a group $G$ is termed a quasiverbal subgroup if ...
1	contained in the kernel of any homomorphism to the quotient	for any homomorphism of groups $\varphi :G\to G/H$, $H$ is contained in the kernel of $\varphi$.
2	intersection of normal subgroups whose quotients satisfy a subgroup-closed group property	there exists a subgroup-closed group property $\alpha$ such that $H$ is the intersection of all normal subgroups $N$ of $G$ for which the quotient group $G/N$ satisfies property $\alpha$.
3	smallest normal subgroup whose quotient satisfies a subgroup-closed group property	there exists a subgroup-closed group property $\alpha$ such that $H$ is the unique smallest normal subgroup of $G$ for which the quotient group $G/H$ satisfies $\alpha$. Note that every $\alpha$ that works for the preceding definition need not work for this definition.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup	union of the images of word maps			|FULL LIST, MORE INFO
pseudoverbal subgroup	intersection of normal subgroups with quotients in a pseudovariety			|FULL LIST, MORE INFO
quasiverbal subgroup	intersection of normal subgroups with quotients in a quasivariety			|FULL LIST, MORE INFO
normal subgroup having no nontrivial homomorphism to its quotient group	no nontrivial homomorphism from the subgroup to the quotient group		|FULL LIST, MORE INFO
normal Sylow subgroup	normal and a Sylow subgroup			|FULL LIST, MORE INFO
Normal Hall subgroup	normal and a Hall subgroup			quotient-subisomorph-containing subgroup|normal Hall subgroup}}

Weaker properties

• Weakly image-closed fully invariant subgroup
• Fully invariant subgroup
• Strictly characteristic subgroup
• Characteristic subgroup

Related properties

• Homomorph-containing subgroup