The notion of braces was introduced by W. Rump [J. Algebra 307, 153–170 (2007; Zbl 1115.16022)] as a generalization of Jacobson radical rings in order to study involutive set-theoretic solutions of the Yang-Baxter equation. Specifically, a (left) brace is a set \(A\) endowed with two binary operations \(+\) and \(\circ\) such that

1.

\((A,+)\) is an abelian group (the additive group of the brace);

2.

\((A,\circ)\) is a group (the multiplicative group of the brace);

3.

\(a\circ (b+c) + a = a\circ b + a\circ c\) holds for all \(a,b,c\in A\) (the brace relation).

Moreover, a brace \(A\) is left nilpotent if \(A^n=0\) for some \(n\in\mathbb{N}\), and right nilpotent if \(A^{(n)}=0\) for some \(n\in\mathbb{N}\). Here \(A^i\) and \(A^{(i)}\) are defined inductively by \[ A = A^1 = A^{(1)},\,\ A^{i+1} = A*A^i,\,\ A^{(i+1)} = A^{(i)} * A, \] where \(*\) is the binary operation \(a * b = a\circ b - a - b\).
The authors noted in the introduction that while several methods exist for characterizing right nilpotent braces of prime power order, braces which are not right nilpotent are more difficult to understand. The paper under review tackles this problem by constructing all braces \(A\) of cardinality \(p^4\) which are not right nilpotent and whose additive group \((A,+)\) is elementary abelian. For \(p>3\), by a result of [D. Bachiller, J. Algebra 453, 160–176 (2016; Zbl 1339.16022)], the multiplicative group \((A,\circ)\) of such a brace must be group XIV or group XV in the notation Burnside. The authors show that group XIV in fact does not occur (Section 8), and they construct all braces \(A\) as described above for which \((A,\circ)\) is group XV (Theorems 5.5 and 6.3). For \(p=2,3\), braces of order \(p^4\) can be calculated the GAP package.