Locally subnormal subgroup

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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

Definition

A subgroup $H$ of a group $G$, is termed locally subnormal if, for every finitely generated subgroup $K$ of $G$, $H$ is a subnormal subgroup of $⟨H,K⟩$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Permutable subgroup		permutable implies locally subnormal		|FULL LIST, MORE INFO
Normal subgroup		(via subnormal)		Subnormal subgroup|FULL LIST, MORE INFO
Subnormal subgroup				|FULL LIST, MORE INFO

References

Textbook references

• Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 216, ^{More info}