具有循环西罗子群的简单斜括号
Simple Skew Braces with Cyclic Sylow Subgroups
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中文总结 AI 辅助
研究有限斜括号在西罗子群循环性假设下的单性与分裂现象，先分类乘法群为\(Z\)-群的有限单斜括号，再考虑更一般情况，证明相关结论并验证拜奥特可解性猜想。
中文摘要 AI 辅助
我们研究在西罗子群循环性假设下有限斜括号中的单性和分裂现象。首先对乘法群为\(Z\)-群的有限单斜括号进行分类。若加法群可解，斜括号要么是素数阶平凡的，要么同构于阶为\(12\)且加法群为\(A_4\)、乘法群为\(C_3\rtimes C_4\)的两个简单斜括号之一；若加法群不可解，则同构于\(\operatorname{PSL}_2(p)\)（\(p\geq5\)为素数）。然后考虑仅对应阶的最小素因子\(p\)的西罗子群为循环的更一般情况，在合适假设下证明斜括号包含一个霍尔\(p'\)-理想并分解为该理想与西罗\(p\)-子括号的半直积。由此得出满足这些假设之一的有限单斜括号是素数阶平凡的。还验证了加法群有循环西罗\(2\)-子群的有限斜括号的拜奥特可解性猜想。
英文摘要
We study simplicity and splitting phenomena in finite skew braces under cyclicity assumptions on Sylow subgroups. We first classify finite simple skew braces whose multiplicative group is a \(Z\)-group. If the additive group is soluble, then the skew brace is either trivial of prime order or isomorphic to one of the two simple skew braces of order \(12\) with additive group \(A_4\) and multiplicative group \(C_3\rtimes C_4\). If the additive group is insoluble, then it is necessarily isomorphic to \(\operatorname{PSL}_2(p)\) for some prime \(p\geq5\). This conclusion is sharp, since such examples exist for every prime \(p\geq5\). We then consider the more general situation in which only a Sylow subgroup corresponding to the smallest prime divisor \(p\) of the order is assumed to be cyclic. Under suitable hypotheses on the additive or multiplicative Sylow \(p\)-subgroup, we prove that the skew brace contains a Hall \(p'\)-ideal and splits as a semidirect product of this ideal with a Sylow \(p\)-subbrace. As a consequence, every finite simple skew brace satisfying one of these hypotheses is trivial of prime order. Moreover, as a consequence of our splitting theorem, we verify Byott's solvability conjecture for finite skew braces whose additive group has a cyclic Sylow $2$-subgroup.
