AI 中文总结
该数学研究证明无限群必存在逆模糊函数，逆模糊自同构仅存在于阿贝尔群，明确有限生成阿贝尔群存在此类自同构的条件，还揭示局部有限群、剩余有限群的阿贝尔性与逆模糊自同构的关联。
AI 中文摘要
设$G$为群，双射$f\colon G\to G$称为逆模糊的，当且仅当对任意$x\in G$，有$f^{-1}(x)=f(x)^{-1}$。我们证明每个无限群都存在逆模糊函数；然而，逆模糊自同构仅能在阿贝尔群上出现。对于有限生成阿贝尔群$A\cong\mathbb{Z}^r\oplus T$（其中$T$有限），我们证明$A$存在逆模糊自同构当且仅当$r$为偶数且$T$存在逆模糊自同构，结合Toborg的有限分类结果可得到明确分类。我们还证明：若无限局部有限群的每个真子群都存在逆模糊自同构，则该群为阿贝尔群，且给出例子说明两个假设均必要。最后，我们证明：若剩余有限群的所有有限商群都存在逆模糊自同构，则该群为阿贝尔群；我们构造了一个无限剩余有限阿贝尔群，其所有有限商群都存在逆模糊自同构，但该群本身不存在逆模糊自同构。
英文摘要
Let $G$ be a group. A bijection $f\colon G\to G$ is called inverse ambiguous if $f^{-1}(x)=f(x)^{-1}$ for every $x\in G$. We prove that every infinite group admits an inverse ambiguous function. An inverse ambiguous automorphism can exist only on an abelian group. We classify the finitely generated and divisible abelian groups admitting such automorphisms. If $A\cong\Z^r\oplus T$, where $T$ is finite, then $A$ admits an inverse ambiguous automorphism if and only if $r$ is even and $T$ admits one. Together with Toborg's finite classification, this gives an explicit classification of the finitely generated abelian groups admitting such automorphisms. For a divisible abelian group $A\cong\mathbb Q^{(κ_0)}\oplus \bigoplus_p C_{p^\infty}^{(κ_p)}$, such an automorphism exists if and only if none of $κ_0$, $κ_2$, or $κ_p$ for $p\equiv3\pmod4$ is a finite odd cardinal. We also prove that if every proper subgroup of an infinite locally finite group $G$ admits an inverse ambiguous automorphism, then $G$ admits one that leaves every subgroup invariant. Finally, we prove that a residually finite group whose finite quotients admit inverse ambiguous automorphisms is abelian. However, we prove the existence of infinite residually finite abelian groups whose finite quotients all admit such automorphisms, but the group itself does not.
