Nearly normal subgroup
From Groupprops
This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: subgroup of finite index and normal subgroup
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Definition
Symbol-free definition
A subgroup of a group is said to be nearly normal if it satisfies the following equivalent conditions:
- It has finite index in its normal closure.
- It is a subgroup of finite index of a normal subgroup of the whole group.
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.
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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 | ||||
subgroup of finite index |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
conjugate-commensurable subgroup | ||||
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 nearly normal if and only if the derived subgroup is finite, viz., the group is a commutator-finite group.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | No | It is possible to have with nearly normal in and nearly normal in , but is not nearly normal in . | |
trim subgroup property | Yes | Every group is nearly normal in itself; the trivial subgroup is nearly normal in every group. | |
intermediate subgroup condition | Yes | nearly normal satisfies intermediate subgroup condition | If and is nearly normal in , then is nearly normal in . |
transfer condition | Yes | nearly normal satisfies transfer condition | If with nearly normal in , then is nearly normal in . |
inverse image condition | Yes | nearly normal satisfies inverse image condition | If is nearly normal and is a homomorphism, then is nearly normal in . |
image condition | Yes | nearly normal satisfies image condition | If is nearly normal and is a surjective homomorphism, then is nearly normal in . |
finite-intersection-closed subgroup property | Yes | nearly normal is finite-intersection-closed | If are nearly normal subgroups of , then is also a nearly normal subgroup. |
finite-join-closed subgroup property | Yes | nearly normal is finite-join-closed | If are nearly normal subgroups of , then is also a nearly normal subgroup. |
conjugate-join-closed subgroup property | Yes | nearly normal is conjugate-join-closed | A join of any number of conjugates of a nearly normal subgroup of a group is nearly normal. |
References
- Groups with finite classes of conjugate subgroups by B.H. Neumann, Math. Z., 63, 1955, Pages 76-96