逆Baer形变与幂零型有限单斜辫子
Inverse Baer deformations and finite simple skew braces of nilpotent type
- University of Cagliari（卡利亚里大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入逆Baer形变，从阿贝尔型辫子构造出加法幂零类至多为二的斜辫子，并利用非对称积的简单左辫子，对每对满足条件的奇素数构造了无穷多个有限单斜辫子，其加法群为非阿贝尔幂零类二群。
AI中文摘要：
具有非阿贝尔加法群的有限单斜辫子在很大程度上仍未得到探索。特别是，据我们所知，此前尚未构造出具有非阿贝尔幂零加法群的有限单斜辫子。我们引入了一种逆Baer形变，它从阿贝尔型辫子产生加法幂零类至多为二的斜辫子。从奇数阶的有限左辫子$C=(A,\oplus,\circ)$和一个合适的双加性交错映射$\kappa:A\times A\to A$出发，我们定义\\[ x+y=x\oplus y\oplus\frac12\kappa(x,y). \\] 所得的斜辫子$\IB_\kappa(C)$满足$\Br(\IB_\kappa(C))=C$且$[x,y]_+=\kappa(x,y)$，并且简单性从$C$传递到$\IB_\kappa(C)$。将此构造应用于由非对称积产生的简单左辫子，对于每对奇素数$p,q$（满足$p\mid(q-1)$）和每个$m\ge1$，我们构造了一个有限单斜辫子$X_{p,q}^{(m)}$，其阶为$p^{2m(q-1)+1}q$，使得\\[ (X_{p,q}^{(m)},+)\cong E_{p,q}^{(m)}\times C_q, \\] 其中$E_{p,q}^{(m)}$是阶为$p^{2m(q-1)+1}$、指数为$p$的超特殊$p$-群。因此，每个固定的可容许对$(p,q)$产生无穷多个两两不同构的有限单斜辫子，其加法群为非阿贝尔幂零类二群。
英文摘要：
Finite simple skew braces with non-abelian additive group remain largely unexplored. In particular, to the best of our knowledge, no finite simple skew brace with non-abelian nilpotent additive group had previously been constructed. We introduce an inverse Baer deformation which produces skew braces of additive nilpotency class at most two from braces of abelian type. Starting from a finite left brace $C=(A,\oplus,\circ)$ of odd order and a suitable biadditive alternating map $κ:A\times A\to A$, we define \[ x+y=x\oplus y\oplus\frac12κ(x,y). \] The resulting skew brace $\IB_κ(C)$ satisfies $\Br(\IB_κ(C))=C$ and $[x,y]_+=κ(x,y)$, and simplicity passes from $C$ to $\IB_κ(C)$. Applying this construction to simple left braces arising from asymmetric products, for every pair of odd primes $p,q$ with $p\mid(q-1)$ and every $m\ge1$ we construct a finite simple skew brace $X_{p,q}^{(m)}$ of order $p^{2m(q-1)+1}q$ such that \[ (X_{p,q}^{(m)},+)\cong E_{p,q}^{(m)}\times C_q, \] where $E_{p,q}^{(m)}$ is an extraspecial $p$-group of order $p^{2m(q-1)+1}$ and exponent $p$. Thus every fixed admissible pair $(p,q)$ yields infinitely many pairwise non-isomorphic finite simple skew braces with non-abelian nilpotent additive group of class two.
