We describe the left brace structure of the structure group and the permutation group associated to an involutive, non-degenerate set-theoretic solution of the quantum Yang-Baxter equation by using the Cayley graph of its permutation group with respect to its natural generating system. We use our descriptions of the additions in both braces to obtain new properties of the structure and the permutation groups and to recover some known properties of these groups in a more transparent way.

Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.

The main aim of this paper is to prove an orbit theorem and to apply it to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite solvable groups.

In this paper we first investigate the relationship between the absolute center

In this paper, we investigate the differences between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe a class of groups whose weak second maximal subgroups coincide with its second maximal subgroups(called WSM-groups) under the solvable case. As an application, we will prove that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.Mathematics Subject Classification (2010): 20C15, 20D10, 20D30.