Almost normal 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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This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Symbol-free definition

A subgroup of a group is said to be almost normal if it satisfies the following equivalent conditions:

• Its normalizer has finite index in the whole group
• It has finitely many conjugate subgroups

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup				|FULL LIST, MORE INFO
Subgroup of finite index				|FULL LIST, MORE INFO

Related properties

• Nearly normal subgroup

Facts

Every subgroup of a group is almost normal if and only if the center has finite index, or equivalently, if the inner automorphism group of the group is finite.

References

• Groups with finite classes of conjugate subgroups by B.H. Neumann, Math. Z., 63, 1955, Pages 76-96