Almost 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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Definition

A subgroup $H$ of a group $G$ is termed an almost subnormal subgroup (or sometimes f-subnormal subgroup) of $G$ if there exists a chain $H=H_{0}\leq H_{1}\leq H_{2}\dots \leq H_{n}=G$, such that for $0\leq i\leq n-1$, $H_{i}$ is either a normal subgroup or a subgroup of finite index in $H_{i+1}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup				Subnormal subgroup|FULL LIST, MORE INFO
Subnormal subgroup	chain from subgroup to group, each normal in successor			|FULL LIST, MORE INFO
Subgroup of finite index	has finite index in the whole group			|FULL LIST, MORE INFO
Almost normal subgroup	has finitely many conjugate subgroups			|FULL LIST, MORE INFO
Nearly normal subgroup	has finite index in its normal closure			|FULL LIST, MORE INFO