Nearly normal subgroup

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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:

  1. It has finite index in its normal closure.
  2. 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.
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
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

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
Transitive subgroup property No It is possible to have H \le K \le G with H nearly normal in K and K nearly normal in G, but H is not nearly normal in G.
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 H \le K \le G and H is nearly normal in G, then H is nearly normal in K.
Transfer condition Yes Nearly normal satisfies transfer condition If H, K \le G with H nearly normal in G, then H \cap K is nearly normal in K.
Inverse image condition Yes Nearly normal satisfies inverse image condition If H \le G is nearly normal and \varphi:M \to G is a homomorphism, then \varphi^{-1}(H) is nearly normal in M.
Image condition Yes Nearly normal satisfies image condition If H \le G is nearly normal and \varphi:G \to M is a surjective homomorphism, then \varphi(H) is nearly normal in M.
Finite-intersection-closed subgroup property Yes Nearly normal is finite-intersection-closed If H,K are nearly normal subgroups of G, then H \cap K is also a nearly normal subgroup.
Finite-join-closed subgroup property Yes Nearly normal is finite-join-closed If H,K are nearly normal subgroups of G, then \langle H, K \rangle 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