Some Braces of Cardinality $p^{4}$ and Related Hopf-Galois Extensions

Abstract

We describe all $\mathbb F_{p}$-braces of cardinality $p^{4}$ which are not right nilpotent. Our solution illustrates a general way of investigating $\mathbb F_{p}$-braces of cardinality $p^{n}$ with a given multiplicative group. The constructed braces are left nilpotent, solvable and prime, and they also contain a non-zero strongly nilpotent ideal.