AI 中文总结
研究有限斜括号乘法群幂零性对其理想结构和可解性的影响，证明相关结论，探讨双边斜括号可解性，得出有限双边斜括号可解性与加或乘法群可解性的关系及无限情况下的相关结论。
AI 中文摘要
我们研究有限斜括号乘法群的幂零假设如何限制其理想结构和可解性。第一个主要结果表明，若\(B=(B,+,\cdot)\)有限且\((B,\cdot)\)幂零，则加法菲廷子群\(F(B,+)\)是\(B\)的非零理想。由此得出，每个具有幂零乘法群的有限单斜括号同构于\(\Triv(C_p)\)，且有素数指数的理想。还表明乘法群幂零一般不蕴含左幂零或可解性。受此阻碍，研究了双边斜括号的可解性，证明了内部换位子理想\([I,I]_I\)是理想，得出可解性的扩展定理，对有限双边斜括号，其可解性等价于加法或乘法群可解，无限双边斜括号虽一般不等价，但加法群可解时，乘法群的每个有限同态像可解。
英文摘要
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.
Comments18 pages
