We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

中文翻译：

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部分自同态、自由合并和自同态群的相干扩展

对于有限结构的部分自同构，我们给出了 Herwig-Lascar 和 Hodkinson-Otto 扩展定理的强化版本。这种强化对同质结构产生了几种组合和群论结果。例如，我们为某些 Fraïssé 类的部分自同构建立了一种连贯形式的扩展属性。我们从这些结果推导出有理 Urysohn 空间的等距群，任何 Fraïssé 类的 Fraïssé 极限的自同构群${\cal F}$- 自由结构（在 Herwig-Lascar 意义上），以及有限关系语言上任何自由齐次结构的自同构群都包含一个稠密的局部有限子群。我们还表明，任何自由的同构结构都允许大量泛型。