Pushforwardable equals inner
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 |
- is a pushforwardable automorphism of : For any homomorphism , there exists an automorphism of such that .
- is an inner automorphism.
An automorphism of a group is termed a pushforwardable automorphism if, for any homomorphism , there exists an automorphism of such that .
Further information: Inner automorphism
- 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 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.