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Nilpotency concepts for skew braces are among the main tools with which we are nowadays classifying certain special solutions of the Yang–Baxter equation, a consistency equation that plays a relevant role in quantum statistical mechanics and in many areas of mathematics. In this context, two relevant questions have been raised in F. Cedó, A. Smoktunowicz and L. Vendramin (Skew left braces of nilpotent type. Proc. Lond. Math. Soc. (3) 118 (2019), 1367–1392) (see questions 2.34 and 2.35) concerning right- and central nilpotency. The aim of this short note is to give a negative answer to both questions: thus, we show that a finite strong-nil brace B need not be right-nilpotent. On a positive note, we show that there is one (and only one, by our examples) special case of the previous questions that actually holds. In fact, we show that if B is a skew brace of nilpotent type and 
$b\ \ast \ b=0$ for all 
$b\in B$, then B is centrally nilpotent.
The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if 
$(B,+)$ is nilpotent.
Given a finite presentation of the structure skew brace 
$G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if 
$G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.
A group 
$G=AB$ is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing 
$A \cap B$ and B permutes with every subgroup of A containing 
$A \cap B$. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra 595 (2022), 434–443] who showed that if 
$G'$ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, then 
$G^{\mathfrak {F}}=A^{\mathfrak {F}}B^{\mathfrak {F}} $, where 
$ \mathfrak {F} $ is a saturated formation containing 
$ \mathfrak {U} $, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning 
$ \mathfrak {F} $-residuals, 
$ \mathfrak {F} $-projectors and 
$\mathfrak {F}$-normalisers. As an application of some of our arguments, we unify some results on weakly mutually 
$sn$-products.
A subgroup A of a group G is said to be hereditarily G-permutable with a subgroup B of G, if 
$AB^x = B^xA$ for some element 
$x \in \langle A, B \rangle $. A subgroup A of a group G is said to be hereditarily G-permutable in G if A is hereditarily G-permutable with every subgroup of G. In this paper, we investigate the structure of a finite group G with all its Schmidt subgroups hereditarily G-permutable.
A classical result of Baer states that a finite group G which is the product of two normal supersoluble subgroups is supersoluble if and only if Gʹ is nilpotent. In this article, we show that if G = AB is the product of supersoluble (respectively, w-supersoluble) subgroups A and B, A is normal in G and B permutes with every maximal subgroup of each Sylow subgroup of A, then G is supersoluble (respectively, w-supersoluble), provided that Gʹ is nilpotent. We also investigate products of subgroups defined above when 
$ A\cap B=1 $ and obtain more general results.
In this paper, we study the structure of finite groups 
$G=AB$
which are a weakly mutually 
$sn$
-permutable product of the subgroups A and B, that is, A permutes with every subnormal subgroup of B containing 
$A \cap B$
and B permutes with every subnormal subgroup of A containing 
$A \cap B$
. We obtain generalisations of known results on mutually 
$sn$
-permutable products.
Assume that 
$G$ is a finite group and 
$H$ is a 2-nilpotent Sylow tower Hall subgroup of 
$G$ such that if 
$x$ and 
$y$ are 
$G$-conjugate elements of 
$H\cap G^{\prime }$ of prime order or order 4, then 
$x$ and 
$y$ are 
$H$-conjugate. We prove that there exists a normal subgroup 
$N$ of 
$G$ such that 
$G=HN$ and 
$H\cap N=1$.
For a formation 
$\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉
$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an 
$fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.
A subgroup 
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ of a finite group 
$G$ is said to be S-semipermutable in 
$G$ if 
$H$ permutes with every Sylow 
$q$-subgroup of 
$G$ for all primes 
$q$ not dividing 
$|H |$. A finite group 
$G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of 
$G$ are S-semipermutable in 
$G$. The aim of the present paper is to characterise the finite MS-groups.
All groups considered in this paper are finite. A subgroup 
$H$ of a group 
$G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of 
$G$ containing 
$H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of 
$G$ has index a power of a prime if and only if 
$G/ \Phi (G)$ is a solvable PST-group. Let 
$\mathfrak{X}$ denote the class of groups 
$G$ all of whose primitive subgroups have prime power index. It is established here that a group 
$G$ is a solvable PST-group if and only if every subgroup of 
$G$ is an 
$\mathfrak{X}$-group.
