Kernel--wreath constructions and infinite families of finite simple skew braces

RESUMO

We introduce a kernel--wreath construction for finite skew braces. Let $C$ be a finite skew brace, write $(C,+)=A$ and $(C,\circ)=R$, and let $\chi:R\twoheadrightarrow T$ be an epimorphism onto a non-trivial finite abelian group. We show that a natural index-$|T|$ kernel in the permutational wreath product $R\wr T$ acts regularly on $A^T$, and hence defines a new skew brace $\KT_T(C,\chi)$ with additive group $A^{|T|}$. The construction preserves every finite abelian quotient of the multiplicative group, as well as solvability, and contains the seed brace $C$ as a diagonal subbrace. It can therefore be iterated indefinitely. More precisely, if $Q$ is a non-trivial finite abelian quotient of $R$ and $p\in\pi(Q)$, then one obtains an infinite tower \[ C=C_0\hookrightarrow C_1\hookrightarrow C_2\hookrightarrow\cdots \] with $(C_m,+)\cong A^{p^m}$ for every $m\geq0$. Our main permanence result shows that if $C$ is simple and is not a trivial skew brace, then $\KT_T(C,\chi)$ is again simple. Consequently, a single simple seed whose multiplicative group has a non-trivial finite abelian quotient gives rise to infinite families of finite simple skew braces. In particular, starting from suitable solvable regular subgroups of the holomorph of a finite non-abelian simple group $S$, we obtain infinite families of simple skew braces with additive groups $S^{p^m}$ and solvable multiplicative groups. Further applications are given to the simple skew braces of order $12$ and to Byott's family.