Let 𝔉 be a saturated formation of soluble Leibniz algebras. Let K be an 𝔉-projector and A/B a chief factor of the soluble Leibniz algebra L. It is well known that if A/B is 𝔉-central, then K covers A/B. I prove the converse: if K covers A/B, then A/B is 𝔉-central.

In Cossey and Stonehewer [‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova125 (2011), 81–105] it is shown that for any odd prime p and integer n≥3, there is a finite p-group G of exponent pn containing a quasinormal subgroup H of exponent pn−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, pn−1 or, when n≥4 , pn−2. Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property 𝔛 of finite p-groups such that (i) 𝔛 is invariant under subgroup lattice isomorphisms and (ii) every chain of 𝔛-subgroups of a finite p-group can be refined to a composition series of 𝔛-subgroups. Failing this, can such a chain always be refined to a series of 𝔛-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest.

Let G be a graph of order n, and let a and b be two integers with 1≤a≤b. Let h:E(G)→[0,1] be a function. If a≤∑ e∋xh(e)≤b holds for any x∈V (G), then we call G[Fh] a fractional [a,b] -factor of G with indicator function h, where Fh ={e∈E(G):h(e)>0}. A graph G is fractional independent-set-deletable [a,b] -factor-critical (in short, fractional ID-[a,b] -factor-critical) if G−I has a fractional [a,b] -factor for every independent set I of G. In this paper, it is proved that if n≥((a+2b)(a+b−2)+1 )/b and δ(G)≥((a+b)n )/(a+2b ) , then G is fractional ID-[a,b] -factor-critical. This result is best possible in some sense, and it is an extension of Chang, Liu and Zhu’s previous result.

A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not.

We consider the class of solvable groups in which all subnormal subgroups have subnormal normalizers, a class containing many well-known classes of solvable groups. Groups of this class have Fitting length three at most; some other information connected with the Fitting series is given.

In this paper, we prove the existence of the ground state for the spinor Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field in the one-dimensional case. We also characterise the ground states of spin-1 Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field; that is, for ferromagnetic systems, we show that, under some condition, searching for the ground state of ferromagnetic spin-1 Bose–Einstein condensates with an external Ioffe–Pitchard magnetic field can be reduced to a ‘one-component’ minimisation problem.

An endomorphism φ of a polynomial ring is said to preserve outer rank if φ sends each polynomial to one with the same outer rank. For the polynomial ring in two variables over a field of characteristic 0 we prove that an endomorphism φ preserving outer rank is an automorphism if one of the following conditions holds: (1) the Jacobian of φ is a nonzero constant; (2) the image of φ contains a coordinate; (3) φ has a ‘fixed point’.

In this paper, solubility of groups factorised as a product of two subgroups which are connected by certain permutability properties is studied.

In this paper we consider the discreteness of spectrum for higher-order differential operators in weighted function spaces. Using the method of embedding theorems of weighted Sobolev spaces Hnp in weighted spaces Ls,r, we obtain a new sufficient and necessary condition to ensure that the spectrum is discrete, which can be easily used to judge the discreteness of some differential operators.

Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups of G with maximal Chermak–Delgado measure, denoted 𝒞𝒟(G), is a sublattice of the lattice of all subgroups of G. In this paper we note that if H∈𝒞𝒟(G) then H is subnormal in G and prove that if K is a second finite group then 𝒞𝒟(G×K)=𝒞𝒟(G)×𝒞𝒟(K) . We additionally describe the 𝒞𝒟(G≀Cp) where G has a nontrivial centre and p is an odd prime and determine conditions for a wreath product to be a member of its own Chermak–Delgado lattice. We also examine the behaviour of centrally large subgroups, a subset of the Chermak–Delgado lattice.

We consider the Brocard–Ramanujan type Diophantine equation y2=x!+A and ask about values of A∈ℤ for which there are at least three solutions in the positive integers. In particular, we prove that the set 𝒜 consisting of integers with this property is infinite. In fact we construct a two-parameter family of integers contained in 𝒜. We also give some computational results related to this equation.

In this paper we study impulsive periodic solutions for second-order nonautonomous singular differential equations. Our proof is based on the mountain pass theorem. Some recent results in the literature are extended.

We characterise the strong- and weak-type boundedness of the geometric fractional maximal operator between weighted Lebesgue spaces in the case 0<p≤q<∞, generalising and improving some older results.

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.

Let k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

Let G be a finite d-dimensional classical group and p a prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2 and either act irreducibly on X or preserve a particular decomposition of X into two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion in G of these so-called p-abundant elements is at least an absolute constant multiple of the best currently known lower bound for the proportion of all p-singular elements. From a computational point of view, the p-abundant elements generalise another class of p-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope that p-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element is p-abundant, both for a known prime p and for the case where p is not known a priori.

Let ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.

We prove that the sequence {log ζ(n)}n≥2 is not holonomic, that is, does not satisfy a finite recurrence relation with polynomial coefficients. A similar result holds for L-functions. We then prove a result concerning the number of distinct prime factors of the sequence of numerators of even indexed Bernoulli numbers.

Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

Let 𝒯n(F) denote the monoid of all upper triangular n×n matrices over a finite field F. It has been shown by Volkov and Goldberg that 𝒯n(F) is nonfinitely based if ∣F∣>2 and n≥4, but the cases when ∣F∣>2 and n=2,3 or when ∣F∣=2 have remained open. In this paper, it is shown that the monoid 𝒯2 (F) is finitely based when ∣F∣=2 , and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by 𝒯2 (F) with ∣F∣=2 are determined.