AI 中文总结
构造有限斜括号，加法群完美，乘法群非完美且几乎单，对《库罗夫卡笔记本》问题20.109作答。结合特定加法群与乘法群的斜括号、自同构扩张分裂及半直积来构造，加法群为\(\PSU_5(64)\times A_5\)，乘法群为\(\Aut(\PSU_5(64))\)。
AI 中文摘要
我们构造了一个有限斜括号，其加法群是完美的，乘法群是非完美且几乎单的。这对《库罗夫卡笔记本》第二十一版中的问题20.109给出了肯定答案。我们例子的加法群和乘法群分别是\(\PSU_5(64)\times A_5\)和\(\Aut(\PSU_5(64))\)。该构造结合了加法群为\(A_5\)、乘法群为\((C_5\rtimes C_4)\times C_3\)的斜括号、\(\PSU_5(64)\)自同构扩张的分裂以及斜括号的半直积。
英文摘要
We construct a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. This gives an affirmative answer to Problem 20.109 in the twenty-first edition of the Kourovka Notebook. The additive and multiplicative groups of our example are \[ \PSU_5(64)\times A_5 \quad\text{and}\quad \Aut(\PSU_5(64)), \] respectively. The construction combines a skew brace with additive group \(A_5\) and multiplicative group \((C_5\rtimes C_4)\times C_3\), the splitting of the automorphism extension of \(\PSU_5(64)\), and a semidirect product of skew braces.
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