Diagram-extensible automorphism

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is termed diagram-extensible if the following holds. Given the following data:


 * A partially ordered set $$I$$ with unique minimum element $$0$$
 * A group $$G_i$$ for each $$i \in I$$ such that $$G_0 = G$$
 * Inclusions $$G_i \to G_j$$ for $$i < j$$, compatible with composition

There exist automorphisms $$\sigma_i$$ of $$G_i$$ for each $$i$$ such that whenever $$i < j$$ the restriction of $$\sigma_j$$ to $$G_i$$ is $$\sigma_i$$, and such that $$\sigma_0 = \sigma$$.

Stronger properties

 * Inner automorphism

Weaker properties

 * Chain-extensible automorphism
 * 2-glue-extensible automorphism
 * Ascending chain-extensible automorphism
 * Finite-chain-extensible automorphism