Inner automorphism to automorphism is right tight for normality
Contents
Statement
Property-theoretic statement
Consider the subgroup property of being a Normal subgroup (?). Then, the following function restriction expression for this subgroup property:
Inner automorphism Automorphism
is a right tight function restriction expression.
Statement with symbols
Suppose is a group and
is a function. Consider the following property for
:
There exists a group containing
as a normal subgroup, and an inner automorphism
of
such that the restriction of
of
.
This property is precisely equivalent to being an automorphism. (Note that by definition of normality, any such
must be an automorphism, so the content of the theorem is that every automorphism can be realized this way).
Facts used
- Every group is normal fully normalized in its holomorph: If
is a group, there is a group
(the holomorph of
) containing
as a normal subgroup, with the property that every automorphism of
extends to an inner automorphism in
.
Related facts
Other facts using this idea or similar proofs
- Left transiter of normal is characteristic: If
is such that whenever
is normal in
,
is also normal in
, then
is characteristic in
. This fact follows from the statement on this page and the right tightness theorem.
- Every group is normal fully normalized in its holomorph
- Normal upper-hook fully normalized implies characteristic
If we drop the requirement that be normal in
, then the result is no longer true. In fact, we have:
- Every injective endomorphism arises as the restriction of an inner automorphism
- Isomorphic iff potentially conjugate: Two subgroups of a group that are isomorphic are conjugate in some bigger group containing it.
- Isomorphic iff potentially conjugate in finite: Two subgroups of a finite group that are isomorphic are conjugate in some bigger finite group containing it.
Proof
Proof using the holomorph
The proof using the holomorph finds a single big group that works for all automorphisms.
Given: A group , an automorphism
of
.
To prove: There exists a group containing
as a normal subgroup, such that
extends to an inner automorphism of
.
Proof: Let be the holomorph of
. Fact (1) now completes the proof.
Proof using semidirect product with just that automorphism
This proof uses a more minimalistic construction, but the construction differs for every automorphism. This proof technique generalizes to proving that every injective endomorphism arises as the restriction of an inner automorphism.
Given: A group , an automorphism
of
.
To prove: There exists a group containing
as a normal subgroup, such that
extends to an inner automorphism of
.
Proof: Consider the group , where the generator of
acts on
by
.
is normal in the semidirect product by definition, and the automorphism
extends to the inner automorphism of
induced by conjugation by that generator of
.