Between normal and characteristic and beyond

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), contrasting subnormality of various depths, and subnormal-to-normal and normal-to-characteristic.

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.

Definitions

Normal subgroup

Further information: normal subgroup

A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

• It is invariant under all inner automorphisms. Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).
• It is the kernel of a homomorphism from the group.
• It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
• Its left cosets are the same as its right cosets (that is, it commutes with every element of the group)

Characteristic subgroup

A subgroup of a group is termed characteristic if it satisfies the following equivalent conditions:

• Every automorphism of the whole group takes the subgroup to within itself
• Every automorphism of the group restricts to an endomorphism of the subgroup
• Every automorphism of the group restricts to an automorphism of the subgroup

Relation between normality and characteristicity

Further information: Normal versus characteristic

We have the following basic implication relations:

• Characteristic implies normal: A characteristic subgroup must be normal, since invariance under all automorphisms implies invariance under inner automorphisms.
• Normal not implies characteristic: A normal subgroup need not be characteristic. For instance, in the group $G \times G$, both factors are normal but the coordinate exchange automorphism interchanges them, so neither is characteristic.

The transiter relation

• Normality is not transitive: A normal subgroup of a normal subgroup need not be normal.
• Characteristic of normal implies normal
• Left transiter of normal is characteristic: If $H \le K$ is a subgroup such that whenever $K$ is normal in a group $G$, so is $H$, then $H$ is characteristic in $K$.

The metaproperties satisfied and not satisfied

Further information: Normal versus characteristic

• Normality satisfies intermediate subgroup condition, while characteristicity does not satisfy intermediate subgroup condition: If $H \le K \le G$, and $H$ is normal in $G$, $H$ is normal in $K$. The analogous statement fails for characteristic subgroups.
• Normality satisfies image condition, while characteristicity does not satisfy image condition: The image of a normal subgroup under a surjective homomorphism is normal, but the image of a characteristic subgroup under a surjective homomorphism need not be characteristic.

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 between them is to look for invariance properties with respect to automorphism properties that are weaker than being an inner automorphism.

If $\alpha$ is an automorphism property such that every inner automorphism of a group satisfies $\alpha$, then the property of being an $\alpha$-invariant subgroup is stronger than normality and weaker than characteristicity.

Automorphisms of certain orders

Further information: cofactorial automorphism-invariant subgroup, p-automorphism-invariant subgroup

Suppose $G$ is a finite group. Then, $\operatorname{Inn}(G) \cong G/Z(G)$, 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 $\operatorname{Aut}(G)$ whose order has no prime factors other than those of $G$. In other words, if $\pi$ is the set of prime factors of the order of $G$, we are looking for the subgroup of $\operatorname{Aut}(G)$ generated by all the $\pi$-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.