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.

Let $A$ be a subgroup of a finite group $G$ and $\sum \,=\,\{{{G}_{0}}\,\le \,{{G}_{1}}\,\le \,.\,.\,.\,\le \,{{G}_{n}}\}$ some subgroup series of $G$. Suppose that for each pair $\left( K,\,H \right)$ such that $K$ is a maximal subgroup of $H$ and ${{G}_{i-1}}\,\le \,K\,<\,H\,\le \,{{G}_{i}}$, for some i, either $A\,\cap \,H\,=\,A\,\cap \,K\,\text{or}\,\text{AH}\,\text{=}\,\text{AK}$. Then $A$ is said to be $\sum$-embedded in $G$. And $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and $a\,\{1\,\le \,G\}$-embedded subgroup $C$ in $G$ such that $G\,=\,AT$ and $T\cap A\,\le \,C\,\le \,A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order ${{p}^{k}}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.

A subgroup H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup Hse of G contained in H such that G=HT and H∩T≤Hse. In this note, we study the influence of the weakly s-permutably embedded property of subgroups on the structure of G, and obtain the following theorem. Let ℱ be a saturated formation containing 𝒰, the class of all supersolvable groups, and G a group with E as a normal subgroup of G such that G/E∈ℱ. Suppose that P has a subgroup D such that 1<∣D∣<∣P∣ and all subgroups H of P with order ∣H∣=∣D∣ are s-permutably embedded in G. Also, when p=2 and ∣D∣=2 , we suppose that each cyclic subgroup of P of order four is weakly s-permutably embedded in G. Then G∈ℱ.