Class-preserving not implies inner

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., class-preserving automorphism) need not satisfy the second automorphism property (i.e., inner automorphism)
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Statement

We can find a group with a class-preserving automorphism (an automorphism sending every element to within its conjugacy class) that is not an inner automorphism (i.e., there does not exist any single element of the group conjugation by which equals the automorphism.

In symbols, we can find a group $G$ and an automorphism $\sigma$ of $G$ such that:

• For every $x\in G$, there exists $g$ (dependent on $x$) such that $\sigma (x)=gxg^{-1}$
• There does not exist any single $g\in G$ satisfying $\sigma (x)=gxg^{-1}$ for all $x\in G$.

Partial truth

Groups where the implication holds

Further information: Group in which every class-preserving automorphism is inner

There are certain important classes of groups where every class-preserving automorphism is inner. For instance, all the symmetric groups on finite sets have this property, as do certain groups arising in differential topology.

Related facts

Stronger facts

• Class-preserving not implies subgroup-conjugating

An equivalent fact

The existence of class-preserving automorphisms that are not inner is closely related to the following fact: conjugacy-closed normal not implies central factor. In other words, a conjugacy-closed normal subgroup (a normal subgroup that is also conjugacy-closed: elements in the subgroup that are conjugate in the whole group must be conjugate in the subgroup) need not be a central factor (a normal subgroup such that every inner automorphism of the whole group restricts to an inner automorphism of the subgroup).

Facts used

1. Conjugacy-closed normal not implies central factor

Proof

Proof based on fact (1)

Note that:

• A conjugacy-closed normal subgroup is a subgroup with the property that every inner automorphism of the whole group restricts to a class-preserving automorphism of the subgroup.
• A central factor is a subgroup with the property that every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.

Fact (1) states that there exist conjugacy-closed normal subgroups that are not central factors. This implies that these subgroups have class-preserving automorphisms that are not inner.

An infinite group example

Further information: Finitary symmetric group is conjugacy-closed in symmetric group

One example of this is the finitary symmetric group on an infinite set: the group of all permutations that move only finitely many elements. Consider the automorphism on this group induced via conjugation by an infinitary permutation (a permutation that moves infinitely many elements). This automorphism sends every element to an element in the same conjugacy class, but is not an inner automorphism.

A finite group example

Constructing an example involving a finite group is somewhat more tricky. One construction is as follows. Consider the ring $\mathbb {Z} /8\mathbb {Z}$. Let $A$ be the additive group of this ring, and $G$ the multiplicative group of units. Let $E$ be the semidirect product $A\rtimes G$.

Now, we use three facts:

• The stability group of $1\triangleleft A\triangleleft E$ corresponds to the elements that are 1-cocycles for the action of $G$ on $A$
• The elements in this stability group that come from inner automorphisms, are those that correspond to 1-coboundaries
• The elements in this stability group that send every element to an element in the same conjugacy class, correspond to 1-cocycles such that for each element, it looks like a 1-coboundary. In other words, it is a 1-cocycle $\varphi$ for the action of $G$ on $A$, such that for every $g\in G$, there exists $a\in A$ (depending on $g$) such that we have:

$\varphi (g)=g.a-a$

To show that there is a class automorphism that is not inner, we basically need to construct such a 1-cocycle that is not a 1-coboundary. The construction, specifically, is:

$\varphi (1)=\varphi (7)=0,\varphi (3)=\varphi (5)=4$

References

Textbook references

• Finite groups with class-preserving outer automorphisms by G. E. Wall, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Page 315 - 320(Year 1947): ^{More info}