# Diagram-extensible automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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This term is related to: Extensible automorphisms problem
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## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### 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 $\sigma_0 = \sigma$.