# Almost subnormal subgroup

From Groupprops

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]

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

## Definition

A subgroup of a group is termed an **almost subnormal subgroup** (or sometimes **f-subnormal subgroup**) of if there exists a chain , such that for , is either a normal subgroup or a subgroup of finite index in .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Normal subgroup | Almost normal subgroup, Join-transitively almost subnormal subgroup, Nearly 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 | Almost normal subgroup, Join-transitively almost subnormal subgroup, Nearly normal subgroup|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 |