Almost normal subgroup
From Groupprops
This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and subgroup of finite index
View other such compositions|View all subgroup properties
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 is a normal subgroup of a subgroup of finite index in the whole group.
- It has only finitely many conjugate subgroups.
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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 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 normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup
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 |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Subgroup for which any join of conjugates is a join of finitely many conjugates | ||||
Almost subnormal subgroup |
Related properties
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.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
Transitive subgroup property | No | It is possible to have with almost normal in and almost normal in , but is not almost normal in . | |
Trim subgroup property | Yes | Every group is almost normal in itself; the trivial subgroup is almost normal in every group. | |
Intermediate subgroup condition | Yes | Almost normal satisfies intermediate subgroup condition | If and is almost normal in , then is almost normal in . |
Transfer condition | Yes | Almost normal satisfies transfer condition | If with almost normal in , then is almost normal in . |
Inverse image condition | Yes | Almost normal satisfies inverse image condition | If is almost normal and is a homomorphism, then is almost normal in . |
Image condition | Yes | Almost normal satisfies image condition | If is almost normal and is a surjective homomorphism, then is almost normal in . |
Finite-intersection-closed subgroup property | Yes | Almost normal is finite-intersection-closed | If are almost normal subgroups of , then is also an almost normal subgroup. |
Finite-join-closed subgroup property | Yes | Almost normal is finite-join-closed | If are almost normal subgroups of , then is also an almost normal subgroup. |
Conjugate-join-closed subgroup property | Yes | Almost normal is conjugate-join-closed | A join of any number of conjugates of an almost normal subgroup of a group is almost normal. |
References
- Groups with finite classes of conjugate subgroups by B.H. Neumann, Math. Z., 63, 1955, Pages 76-96