# Quotient-subisomorph-containing subgroup

## Contents

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

## 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 Pseudoverbal subgroup, Quasiverbal subgroup|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 Normal subgroup having no nontrivial homomorphism to its quotient group|FULL LIST, MORE INFO
Normal Hall subgroup normal and a Hall subgroup quotient-subisomorph-containing subgroup|normal Hall subgroup}}