The Group Properties Wiki (pre-alpha)
TIP: Beware of terminology local to the wiki
ABOUT US: We use Semantic MediaWiki. Learn more about using Semantic MediaWiki
ALSO CHECK OUT: Diffgeom: The Differential Geometry Wiki
Between normal and characteristic and beyond
From Groupprops
This is a survey article describing notions intermediate between the following two notions: normal subgroup and characteristic subgroup
View other survey articles about normal subgroup | View other survey articles about characteristic subgroup
YOU MAY ALSO BE INTERESTED IN: normal versus characteristic (a comparison of the subgroup properties of normality and characteristicity), varying normality (discusses variations on the subgroup property of normality from a variety of angles), varying characteristicity (discusses variations of the subgroup property of characteristicity from a variety of angles), between normal and subnormal and beyond, and from normal to characteristic and subnormal to normal.
This survey article looks at various subgroup properties that lie somewhere between the property of being a normal subgroup and the property of being a characteristic subgroup. The subgroup properties are organized according to different running themes.
Contents |
Review of the definitions
Fill this in later
Relation between normality and characteristicity
Further information: Normal versus characteristic
The transiter relation
The metaproperties satisfied and not satisfied
One notion of betweenness: invariance under the right kind of automorphisms
Normality is defined as the property of being invariant under all inner automorphisms, while characteristicity is defined as the property of being invariant under all automorphisms. Thus, one way of looking for properties in betweenthem is to look for invariance properties with respect to automorphism properties that are weaker than being an inner automorphism.
If α is an automorphism property such that every inner automorphism of a group satisfies α, then the property of being an α-invariant subgroup is stronger than normality and weaker than characteristicity.
Extensible automorphism
Further information: Extensible automorphism, Inner implies extensible, Extensible implies subgroup-conjugating, Extensible automorphism-invariant equals normal
An automorphism σ of a group G is termed an extensible automorphism if, for any embedding 
in a bigger group H, there exists an automorphism σ' of H such that the restriction of σ' to G is σ.
An automorphism σ of a group G is termed an infinity-extensible automorphism if it can be extended to something that is again infinity-extensible, i.e., if it can be recursively extended.
Inner automorphisms are infinity-extensible, because every inner automorphism of a subgroup extends to an inner automorphism of the whole group.
Thus, we have the following chain of implications:
Inner automorphism 
Infinity-extensible automorphism Extensible automorphism Automorphism
We thus see that the property of a subgroup being invariant under all extensible automorphisms is stronger than the property of being a normal subgroup. It turns out, however, that every extensible automorphism of a group is subgroup-conjugating, and for this reason, the property of being an extensible automorphism-invariant subgroup is equal to the property of being a normal subgroup.
Automorphisms of certain orders
Further information: cofactorial automorphism-invariant subgroup, p-automorphism-invariant subgroup
Suppose G is a finite group. Then, 
, and hence, the order of the inner automorphism group of G divides the order of G. In particular, every inner automorphism of a group has order with no prime factors other than those of the order of G.
We can look at the set of all elements of 
whose order has no prime factors other than those of G. In other words, if π is the set of prime factors of the order of G, we are looking for the subgroup of generated by all the π-automorphisms.
The property of a subgroup being invariant under all such automorphisms is weaker than characteristicity, but stronger than normality. Such a subgroup is termed a cofactorial automorphism-invariant subgroup.
Of particular interest is the situation where G is a p-group. In this case, we are looking at all the p-automorphism-invariant subgroups.
Remedying the intermediate subgroup condition
The issue
Further information: Normality satisfies intermediate subgroup condition, Characteristicity does not satisfy intermediate subgroup condition
If 
are groups and H is a characteristic subgroup of G, then H need not be characteristic in K. On the other hand, if H is normal in G, then H must be normal in K.
Potentially characteristic=
Further information: Potentially characteristic subgroup, Potentially relatively characteristic subgroup, Strongly potentially characteristic subgroup
- Potentially characteristic subgroup: A subgroup H of a group K is termed potentially characteristic in K if there exists a group G containing K such that H is a characteristic subgroup of G.
- Strongly potentially characteristic subgroup: A subgroup H of a group K is termed strongly potentially characteristic in K if there exists a group G containing K such that both H and K are characteristic subgroups of G.
- Potentially relatively characteristic subgroup: A subgroup H of a group K is termed potentially relatively characteristic in K if, for every automorphism σ of K that extends to an automorphism of G, σ(H) = H.
The implication sequence is:
Characteristic Strongly potentially characteristic Potentially characteristic Potentially relatively characteristic Extensible automorphism-invariant
Remedying the image condition
Some properties obtained by composition
Characteristic subgroup of some special type of normal subgroup
| Survey article about | Normal subgroup +, and Characteristic subgroup + |
