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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This is a variation of subnormal subgroup|Find other variations of subnormal subgroup |

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 $H=H_0\le H_1\le H_2\dots \le H_n=G$, such that for $0\le i\le n-1$, $H_i$ 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