Pushforwardable equals inner
From Groupprops
This article gives a proof/explanation of the equivalence of multiple definitions for the term inner automorphism
View a complete list of pages giving proofs of equivalence of definitions
This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |
Contents
Statement
The following are equivalent for an automorphism of a group
:
-
is a pushforwardable automorphism of
: For any homomorphism
, there exists an automorphism
of
such that
.
-
is an inner automorphism.
Definitions used
Pushforwardable automorphism
An automorphism of a group
is termed a pushforwardable automorphism if, for any homomorphism
, there exists an automorphism
of
such that
.
Inner automorphism
Further information: Inner automorphism
Facts used
- Extensible equals inner: An automorphism
of a group
is inner if and only if it can be extended to an automorphism for any group containing
.
Proof
Proof that inner implies pushforwardable
If for
, and
is a homomorphism, we can take
.
Proof that pushforwardable implies inner
If is pushforwardable, then, in particular,
is extensible in the sense that it can be extended to an automorphism for any group containing
. Thus, by fact (1),
is inner.