Transitively normal subgroup

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

Equivalent definitions in tabular format

No. Shorthand A subgroup of a group is termed transitively normal if ... A subgroup H of a group G is termed transitively normal if ...
1 transitively normal every normal subgroup of the subgroup is normal in the whole group. whenever K is a normal subgroup of H, K is also a normal subgroup of G.
2 normal automorphisms restriction every normal automorphism of the whole group restricts to a normal automorphism of the subgroup. for any normal automorphism \sigma of G, the restriction of \sigma to H is a normal automorphism of H.
3 normal CEP-subgroup it is a normal subgroup as well as a CEP-subgroup: every normal subgroup of it is an intersection with it of a normal subgroup of the whole group H is a normal subgroup of G and for any normal subgroup K of H there exists a normal subgroup L of G such that K = H \cap L.

Equivalence of definitions

Further information: Equivalence of definitions of transitively 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]
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and CEP-subgroup
View other subgroup property conjunctions | view all subgroup properties
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality
CAUTIONARY NOTE: There is a paper where the term transitively normal is used for what we call intermediately subnormal-to-normal subgroup

Examples

Extreme examples

  • Every group is transitively normal as a subgroup of itself.
  • The trivial subgroup is transitively normal in any group.

Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property transitively normal subgroup.

 Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2
SL(2,3) in GL(2,3)General linear group:GL(2,3)Special linear group:SL(2,3)Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

Below are some examples of a proper nontrivial subgroup that does not satisfy the property transitively normal subgroup.

 Group partSubgroup partQuotient part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
D8 in A6Alternating group:A6Dihedral group:D8
D8 in S4Symmetric group:S4Dihedral group:D8
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of M16M16Cyclic group:Z2
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
S2 in S3Symmetric group:S3Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Twisted S3 in A5Alternating group:A5Symmetric group:S3

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes transitive normality is transitive If H \le K \le G are groups such that H is transitively normal in K and K is transitively normal in G, then H is transitively normal in G.
intermediate subgroup condition Yes transitive normality satisfies intermediate subgroup condition If H \le K \le G and H is transitively normal in G, then H is transitively normal in K.
finite-intersection-closed subgroup property No transitive normality is not finite-intersection-closed It is possible to have a group G and transitively normal subgroups H_1,H_2 of G such that H_1 \cap H_2 is not transitively normal in G.
finite-join-closed subgroup property No transitive normality is not finite-join-closed It is possible to have a group G and transitively normal subgroups H_1,H_2 of G such that the join \langle H_1,H_2\rangle (which is the same as the product H_1H_2) is not transitively normal.
centralizer-closed subgroup property No transitive normality is not centralizer-closed It is possible to have a group G and a transitively normal subgroup H of G such that the centralizer C_G(H) is not transitively normal.
image condition Yes transitive normality satisfies image condition If H is a transitively normal subgroup of G and \varphi:G \to K is a surjective homomorphism, then \varphi(H) is a transitively normal subgroup of K.
quotient-transitive subgroup property No transitive normality is not quotient-transitive It is possible to have groups H \le K \le G such that H is transitively normal in G and K/H is transitively normal in G/H, but K is not transitively normal in G.

Relation with other properties

Conjunction with other properties

Here are some conjunctions with other subgroup properties:

Conjunction Other component of conjunction Intermediate notions Additional comments
characteristic transitively normal subgroup characteristic subgroup |FULL LIST, MORE INFO
c-closed transitively normal subgroup c-closed subgroup (equals its double centralizer) |FULL LIST, MORE INFO

Conjunctions with group properties:

Conjunction Property of the group Intermediate notions Additional comments
cyclic normal subgroup Cyclic group Abelian hereditarily normal subgroup, Hereditarily normal subgroup|FULL LIST, MORE INFO for a cyclic subgroup, being normal is equivalent to being transitively normal.
abelian hereditarily normal subgroup Abelian group Hereditarily normal subgroup|FULL LIST, MORE INFO for an abelian subgroup, being transitively normal is equivalent to being hereditarily normal.
hereditarily normal subgroup Dedekind group (every subgroup is normal) SCAB-subgroup|FULL LIST, MORE INFO
nilpotent transitively normal subgroup Nilpotent group |FULL LIST, MORE INFO


Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
central factor product with centralizer is whole group central factor implies transitively normal transitively normal not implies central factor (see also list of examples) Conjugacy-closed normal subgroup, Locally inner automorphism-balanced subgroup, SCAB-subgroup|FULL LIST, MORE INFO
direct factor factor in an internal direct product direct factor implies transitively normal (also, via central factor) (via central factor) (see also list of examples) Central factor, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Right-quotient-transitively central factor, SCAB-subgroup, Upper join of direct factors|FULL LIST, MORE INFO
cocentral subgroup product with center is whole group (via central factor) (via central factor) Central factor, Conjugacy-closed normal subgroup, Right-quotient-transitively central factor, SCAB-subgroup|FULL LIST, MORE INFO
central subgroup contained in the center (via central factor, via hereditarily normal) (via central factor, via hereditarily normal) Abelian hereditarily normal subgroup, Central factor, Conjugacy-closed normal subgroup, Hereditarily normal subgroup, Join-transitively central factor, SCAB-subgroup|FULL LIST, MORE INFO
locally inner autorphism-balanced subgroup every inner automorphism of the whole group restricts to a locally inner automorphism |FULL LIST, MORE INFO
conjugacy-closed normal subgroup both conjugacy-closed and normal conjugacy-closed normal implies transitively normal transitively normal not implies conjugacy-closed |FULL LIST, MORE INFO
SCAB-subgroup any subgroup-conjugating automorphism of whole group restricts to subgroup-conjugating automorphism of subgroup SCAB implies transitively normal Transitively normal not implies SCAB |FULL LIST, MORE INFO
hereditarily normal subgroup every subgroup of it is normal in whole group SCAB-subgroup|FULL LIST, MORE INFO
cyclic normal subgroup cyclic and normal (via SCAB, via hereditarily normal) (via SCAB, via hereditarily normal) Abelian hereditarily normal subgroup, Hereditarily normal subgroup|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup (by definition) normality is not transitive |FULL LIST, MORE INFO
CEP-subgroup every normal subgroup of it arises as its intersection with a normal subgroup of whole group |FULL LIST, MORE INFO
subgroup whose commutator with any subset is normal its commutator with any subset of whole group is normal in whole group commutator of a transitively normal subgroup and a subset implies normal commutator with any subset is normal not implies transitively normal |FULL LIST, MORE INFO

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression H is a transitively normal subgroup of G if ... This means that transitive normality is ... Additional comments
normal automorphism \to normal automorphism every normal automorphism of G restricts to a normal automorphism of H the balanced subgroup property for normal automorphisms Hence, it is a t.i. subgroup property, both transitive and identity-true
inner automorphism \to normal automorphism every inner automorphism of G restricts to a normal automorphism of H
class-preserving automorphism \to normal automorphism every class-preserving automorphism of G restricts to a normal automorphism of H
subgroup-conjugating automorphism \to normal automorphism every subgroup-conjugating automorphism of G restricts to a normal automorphism of H
strong monomial automorphism \to normal automorphism every strong monomial automorphism of G restricts to a normal automorphism of H