Ideals and Solvability in Skew Braces

Abstract

We investigate how nilpotency assumptions on the multiplicative group of a finite skew brace constrain its ideal structure and solvability. Our first main result shows that, if \(B=(B,+,\cdot)\) is finite and \((B,\cdot)\) is nilpotent, then the additive Fitting subgroup \(F(B,+)\) is a non-zero ideal of \(B\). As consequences, every finite simple skew brace with nilpotent multiplicative group is isomorphic to \(\Triv(C_p)\) for some prime \(p\), and every such skew brace admits an ideal of prime index. In particular, \[ B*B\neq B \qquad\text{and}\qquad \partial(B) \neq B. \] We also show that nilpotency of the multiplicative group does not, in general, imply either left nilpotency or solvability. Motivated by this obstruction, we then study the solvability of two-sided skew braces, both in the finite and in the general setting. We prove that, whenever \(I\) is an ideal of a two-sided skew brace \(B\), the internal commutator ideal \([I,I]_I\) is again an ideal of \(B\). This yields an extension theorem for solvability and implies that, for finite two-sided skew braces, solvability of the skew brace is equivalent to solvability of either the additive or the multiplicative group. In particular, every finite skew brace with abelian multiplicative group is solvable. Finally, we show that, although this equivalence fails in general for infinite two-sided skew braces, a residual form of it still survives: if the additive group is solvable, then every finite homomorphic image of the multiplicative group is solvable.