Finite skew braces with characteristically simple multiplicative group

TL;DR

This paper explores finite skew braces with characteristically simple multiplicative groups, revealing strong structural constraints in certain cases and highlighting asymmetries between additive and multiplicative group properties.

Contribution

It establishes new rigidity results for skew braces with simple multiplicative groups and characterizes conditions under which additive and multiplicative groups are isomorphic or constrained.

Findings

If the skew brace is two-sided with non-abelian characteristically simple multiplicative group, then additive and multiplicative groups are isomorphic.

When both additive and multiplicative groups are characteristically simple with the same number of factors, they are necessarily isomorphic.

For supersolvable additive groups with non-abelian simple groups, strong restrictions apply, including specific group isomorphisms and subgroup structures.

Abstract

We study finite skew braces whose multiplicative group is characteristically simple, namely of the form \(S^n\) for a finite simple group \(S\). Motivated by the strong rigidity phenomena known for skew braces with simple or quasisimple multiplicative group, we investigate to what extent the assumption \((B,\cdot)\cong S^n\) constrains the additive group. We first consider the two-sided case and prove that if \(B\) is a finite two-sided skew brace with non-abelian characteristically simple multiplicative group, then \((B,+) \cong (B,\cdot)\). We then show that a similar rigidity phenomenon persists beyond the two-sided setting: if the additive group is also characteristically simple with the same number of direct factors, say \((B,+)\cong T^n\) for some finite simple group \(T\), then necessarily \((B,+)\cong (B,\cdot)\). Next, we consider the case in which \((B,+)\) is supersolvable and \(S\) is non-abelian, and prove strong restrictions: \(S \cong \PSL_2(7)\), the additive group has a quotient isomorphic to \(S_3\), and it contains a non-trivial elementary abelian \(3\)-subgroup with large centralizer. In particular, if \((B,\cdot)\cong S\times S\), then \((B,+)\) cannot be supersolvable. Finally, we investigate skew braces with characteristically simple additive group and obtain several existence and non-existence results depending on the group-theoretic type of the multiplicative group. These results exhibit a marked asymmetry: while a non-abelian characteristically simple multiplicative group imposes strong rigidity on the skew brace, the corresponding condition on the additive group leads to a substantially more flexible existence theory.