强大乘法群不强制有限 braces 的右幂零性
Powerful multiplicative groups do not force right nilpotence in finite braces
- Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）
- imec-SMIT
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究构造了一类有限左 brace，其乘法群为强大 $p$-群但非右幂零，否定了 Shalev–Smoktunowicz 猜想，还得到了 Yang–Baxter 方程的相关集合论解。
AI中文摘要：
对于每个奇素数 $p$，我们构造了一个阶为 $p^{2p+1}$ 的有限左 brace $A_p$，其加法群为初等阿贝尔群，乘法群 $G_p$ 是强大 $p$-群，但 $A_p$ 不是右幂零的。更精确地说，$G_p'=G_p^p\cong C_p^2$，$\operatorname{cl}(G_p)=2$，$\operatorname{exp}(G_p)=p^2$，且 $\operatorname{Soc}(A_p)=0$。因此，即使对于类为 2、导出子群阶为 $p^2$ 的乘法群，强大性也不强制右幂零性。这种阻碍是显式的：$A_p$ 包含一个三维平凡理想 $T$，满足 $T*A_p=T$，而其左序列是幂零交换代数的理想幂滤过。$T$ 和 $A_p/T$ 均为右幂零，故有限左 brace 的右幂零性在扩张下不封闭。该构造是统一的，源于有限局部交换代数、一个平方零导子和一个不变特征，产生一个正则仿射子群作为半直积上特征的核。它在每个奇特征下否定了 Shalev–Smoktunowicz 猜想，并给出了 Yang–Baxter 方程的有限不可收缩对集合论解，其置换群是类为 2 的强大 $p$-群。
英文摘要：
For every odd prime $p$, we construct a finite left brace $A_p$ of order $p^{2p+1}$ whose additive group is elementary abelian and whose multiplicative group $G_p$ is a powerful $p$-group, but such that $A_p$ is not right nilpotent. More precisely, $G_p'=G_p^p\cong C_p^2, \operatorname{cl}(G_p)=2, \exp(G_p)=p^2,$ and $\operatorname{Soc}(A_p)=0$. Thus powerfulness does not force right nilpotence even for multiplicative groups of class two with derived subgroup of order $p^2$. The obstruction is explicit: $A_p$ contains a three-dimensional trivial ideal $T$ satisfying $T*A_p=T$, whereas its left series is the ideal-power filtration of a nilpotent commutative algebra. Both $T$ and $A_p/T$ are right nilpotent, so right nilpotence of finite left braces is not closed under extensions. The construction is uniform and arises from a finite local commutative algebra, a square-zero derivation, and an invariant group homomorphism, yielding a regular affine subgroup as the kernel of a homomorphism on a semidirect product. It disproves the Shalev--Smoktunowicz conjecture in every odd characteristic and yields finite non-degenerate irretractable involutive set-theoretic solutions of the Yang--Baxter equation whose permutation groups are powerful $p$-groups of class two.
