Soluble skew left braces and soluble solutions of the Yang-Baxter equation (Q6594399)

The paper is concerned with skew left braces. These are sets \(B\) together with two group operations \(+\) and \(\cdot\), constrained by the distributivity condition \N\[\Na\cdot(b+c)=a\cdot b -a + a\cdot c \N\]\Nfor all \(a,b,c\in B\). Note that this implies that the neutral elements of the groups coincide. Skew left braces are of particular interest as they provide set-theoretic solutions of the Yang-Baxter equation and, conversely, set-theoretic solutions of the Yang-Baxter equation induce skew left brace structures on the so-called structure skew left brace. Thus, the classification of set-theoretic solutions of the Yang-Baxter equation is closely related to the classification of skew left braces.\N\NThe paper contributes to the classification of skew left braces by discussing and examining skew left braces with addition structure, primarily soluble skew left braces. This notion of soluble skew left brace is original to this work and differs from previous approaches to soluble structures. The definition is given in terms of ideals and commutators: given an ideal \(I\) of a skew left brace \(B\) we set \(\partial_B(I):=[I,I]_B\), where the latter is a certain ``derived ideal'' commutator. Inductively, this gives a descending sequence of ideals \N\[\NI\supseteq\partial_1(I)\supseteq\partial_2(I)\supseteq\cdots\supseteq\partial_n(I)\supseteq\cdots,\N\]\Nwhere \(\partial_1(I):=\partial_B(I)\) and we define inductively \(\partial_n(I):=\partial_B(\partial_{n-1}(I))\) for all \(n>1\). We call \(I\) soluble if there exists an \(n\) such that \(\partial_n(I)=0\). A skew left brace is called soluble if \(B\subseteq B\) is a soluble ideal and one calls the smallest \(n\) such that \(\partial_n(B)=0\) the derived length of \(B\). For example, abelian skew left braces are soluble braces with derived length \(1\). There is a corresponding notion of soluble set-theoretic solutions of the Yang-Baxter equation and it is proven that these correspond to soluble skew left braces.\N\NIn the main theorem of the paper the authors describe ``chief factors'' of soluble skew left braces \(B\) with chief series. Given ideals \(J\subseteq I\) of \(B\) we call \(I/J\) a chief factor of \(B\) if \(I/J\) is a minimal ideal of \(B/J\), i.e., if \(I/J\neq 0\) and \(0,I/J\) are the only ideals contained in \(I/J\). Furthermore, a chief factor \(I/J\) is said to be a Frattini chief factor of \(B\) if \(I/J\subseteq\Phi(B/J)\), where \(\Phi(B/J)\), the ``Frattini subbrace'', is the intersection of all maximal subbraces of \(B/J\). Moreover, \(I/J\) is called a complemented chief factor of \(B\) if it is complemented in \(B/J\). Then, a series of ideals \N\[\N0=I_0\subseteq I_1\subseteq\cdots\subseteq I_n=B\N\]\Nis called chief series if all \(I_j/I_{j-1}\) are chief ideals. Coming back to the main theorem, each chief factor of a soluble skew left brace \(B\) with chief series is abelian and it is either a Frattini chief factor or a complemented chief factor. In case \(B\) is finite it further follows that every chief factor is isomorphic to an elementary abelian \(p\)-group for some prime \(p\) and each maximal sub skew brace of \(B\) has prime power index as a subgroup of \(B\).\N\NAnother focus of the article is the study of subbraces of finite skew left braces. It is shown that in case a finite skew brace \(B\) does not admit a proper skew subbrace then \(B\) is trivial and isomorphic to a group of prime order.\N\NIn the final chapter of the paper there is a ``working example'', where the authors discuss a skew left brace that is soluble in a weaker sense, using a definition that was already present in the literature. It is proven that this skew left brace is not soluble in the sense of the paper.