Inverse Baer deformations and finite simple skew braces of nilpotent type

Abstract

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.