The Baer Transform for Skew Braces

Abstract

We introduce and study the Baer transform for finite skew braces whose additive group has odd order and nilpotency class at most $2$. Starting from Baer's classical construction for nilpotent groups of odd order and nilpotency class $2$, we replace the additive group $(X,+)$ by an abelian group $(X,\oplus)$ on the same underlying set and prove that \[ Br(X)=(X,\oplus,\cdot) \] is a finite skew brace of abelian type. We show that the Baer transform preserves automorphisms, strong left ideals, ideals and central ideals, and we establish its compatibility with quotients. We then compare structural properties of $X$ and $Br(X)$. As applications, we show that direct product decompositions into ideals are preserved by the Baer transform, giving a criterion for indecomposability of skew braces. Finally, for finite $p$-skew braces with $p$ odd, we apply Thompson critical subgroups together with the Baer transform to embed suitable automorphism groups into automorphism groups of skew braces of abelian type.