Finite-index Problems in Skew Braces

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 \(\lambda\)-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.