A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group

Abstract

We construct a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. This gives an affirmative answer to Problem 20.109 in the twenty-first edition of the Kourovka Notebook. The additive and multiplicative groups of our example are \[ \PSU_5(64)\times A_5 \quad\text{and}\quad \Aut(\PSU_5(64)), \] respectively. The construction combines a skew brace with additive group \(A_5\) and multiplicative group \((C_5\rtimes C_4)\times C_3\), the splitting of the automorphism extension of \(\PSU_5(64)\), and a semidirect product of skew braces.