斜括号中的有限指数问题
Finite-index Problems in Skew Braces
AI总结：
研究斜括号中的有限指数问题，证明了子斜括号的有限加法指数与有限乘法指数的等价性，构造反例回答相关问题，表明特定条件下元素成为\((s)\)-元素，还指出特定单生成自由右幂零斜括号的指数为2的理想的生成特性。
AI中文摘要：
我们研究斜括号中的有限指数问题。对于斜括号\(B\)的每个子斜括号\(A\)，证明了有限加法指数等同于有限乘法指数；当这些指数有限时，它们相等，肯定地回答了文献\cite{CPV}中的问题3.7。接着构造了一个指数为3的左括号及其强左理想，其中不含有限指数理想，否定地回答了文献\cite{CPV}中的问题3.6。还表明有限加法和乘法共轭类以及有限\(\lambda\)-轨道迫使一个元素成为\((s)\)-元素，回答了文献\cite{CPV}中的问题5.21。最后，对于每个\(n\geq 3\)，文献\cite{Free}中引入的\(n\)类单生成自由右幂零斜括号有一个指数为2的理想，它作为斜括号不是有限生成的，尽管作为理想是有限生成的。
英文摘要：
We investigate finite-index problems in skew braces. For every sub-skew brace \(A\) of a skew brace \(B\), we prove that finite additive index is equivalent to finite multiplicative index; whenever these indices are finite, they coincide. This answers Question~3.7 of \cite{CPV} affirmatively. We then construct a left brace with a strong left ideal of index \(3\) containing no finite-index ideal, giving a negative answer to Question~3.6 of \cite{CPV}. We also show that finite additive and multiplicative conjugacy classes, together with a finite \(λ\)-orbit, force an element to be an \((s)\)-element, thereby answering Question~5.21 of \cite{CPV}. Finally, for every \(n\geq 3\), a one-generated free right-nilpotent skew brace of class \(n\), introduced in \cite{Free}, has an index-\(2\) ideal that is not finitely generated as a skew brace, although it is finitely generated as an ideal.
