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19.90 (2018)

Partially Solved

A skew brace is a set $B$ equipped with two operations $+$ and $\cdot$ such that $(B, +)$ is an additively written (but not necessarily abelian) group, $(B, \,\cdot\,)$ is a multiplicatively written group, and $a \cdot (b + c) = ab - a + ac$ for any $a, b, c \in B$.
$\qquad$ a) Is there a skew brace with soluble additive group but non-soluble multiplicative group?
$\qquad$ b) Is there a skew brace with non-soluble additive group but nilpotent multiplicative group?
$\qquad$ c) Is there a finite skew brace with soluble additive group but non-soluble multiplicative group?
$\qquad$ d) Is there a finite skew brace with non-soluble additive group but nilpotent multiplicative group?

Contributor: A. Smoktunowicz, L. Vendramin

Progress

a) Yes, there is (T. Nasybullov, J. Algebra, 540 (2019), 156–167).
d) No, there is not (C. Tsang, Q. Chao, Int. J. Algebra Comput., 30, no. 2 (2020), 253–265).

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