具有弱虚数消去的寡态群的同构问题
The Isomorphism Problem for Oligomorphic Groups with Weak Elimination of Imaginaries
AI总结：
本文证明在弱虚数消去条件下，实数相等关系是寡态群拓扑同构复杂性的严格下界，并将模型论方法用于置换群同构重构及无限维一般线性群自同构描述。
摘要：
文献[21]中提出一个问题：实数上的相等关系是否可作为寡态群之间拓扑同构复杂性的严格下界。我们证明，在弱虚数消去的假设下，情况确实如此。我们的方法是模型论的，并且对从自同构群的（拓扑）同构重构置换群同构这一经典问题也有应用。作为具体应用，我们对有限域上任意维数为 $\aleph_0$ 的向量空间 $V$ 明确描述了 $\mathrm{Aut}(\mathrm{GL}(V))$，这与 Schreier 和 van der Waerden 针对有限维空间给出的经典描述相类似。
英文摘要：
In [21] it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of $\mathrm{Aut}(\mathrm{GL}(V))$ for any vector space $V$ of dimension $\aleph_0$ over a finite field, in affinity with the classical description for finite dimensional spaces due to Schreier and van der Waerden.
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