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

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.

Other closely related articles: 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), and from normal to characteristic and subnormal to normal.

Review of the definitions

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Relation between normality and characteristicity

Further information: Normal versus 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 betweenthem 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.

Extensible automorphism

Further information: Extensible automorphism, Inner implies extensible, Extensible automorphism-invariant subgroup

An automorphism $\sigma$ of a group $G$ is termed an extensible automorphism if, for any embedding $G\le H$ in a bigger group $H$, there exists an automorphism ${\sigma }^{\prime }$ of $H$ such that the restriction of ${\sigma }^{\prime }$ to $G$ is $\sigma$.

An automorphism $\sigma$ 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

This leads to some subgroup properties weaker than characteristicity and stronger than normality:

• Extensible automorphism-invariant subgroup: A subgroup that is invariant under all the extensible automorphisms of the whole group.
• Infinity-extensible automorphism-invariant subgroup: A subgroup that is invariant under all the infinity-extensible automorphisms of the whole group.

The extensible automorphisms conjecture states that every extensible automorphism of a group is inner. If that conjecture were true, then both the above properties would collapse to normality. More generally, if it were true that every extensible automorphism of a group is a normal automorphism, both the above properties would collapse to normality.

Automorphisms of certain orders

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

Suppose $G$ is a finite group. Then, $\mathrm{I}\mathrm{n}\mathrm{n}(G)\cong G/Z(G)$, 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 $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ 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.

Remedying the intermediate subgroup condition

The issue

Further information: Normality satisfies intermediate subgroup condition, Characteristicity does not satisfy intermediate subgroup condition

If $H\le K\le G$ 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 $\sigma$ of $K$ that extends to an automorphism of $G$, $\sigma (H)=H$.

The implication sequence is:

Characteristic $⟹$ Strongly potentially characteristic $⟹$ Potentially characteristic $⟹$ Potentially relatively characteristic $⟹$ Extensible automorphism-invariant

Some properties obtained by composition

Characteristic subgroup of some special type of normal subgroup

• Characteristic subgroup of direct factor
• Characteristic subgroup of central factor