Abstract For a morphism whose domain is a totally disconnected locally compact group, the openness test of Cowling, Hofmann and Morris [‘Open mappings of locally compact groups’, J. Group Theory 27 (6) (2024), 1143–1149] can be stated in terms of compact open subgroups. We use this formulation to study spaces of compact open subgroups. A surjective open morphism with compact kernel induces a quotient map between such spaces, and the inverse-image map identifies the target with a closed retract of the source. Conversely, for a surjective open morphism whose target has a compact open subgroup, compactness of the kernel is forced by the requirement that inverse images of compact open subgroups remain compact.
Abstract A subgroup X of a group G is called normal-by-finite if it is finite over its core X G $X_G$ upper X Subscript upper G . We investigate the behaviour of uncountable groups G of regular cardinality ℵ $\aleph $ normal first transfinite cardinal in which all subgroups of cardinality ℵ $\aleph $ normal first transfinite cardinal are normal-by-finite. Moreover, generalised soluble groups whose subgroups of infinite rank are either normal-by-finite or contranormal are characterised, and the analogous problem is considered for uncountable groups whose subgroups of large cardinality satisfy the same condition.
We introduce and study Liouville number groups, namely additive subgroups of the real numbers in which every nonzero element is a Liouville number. Using continued-fraction methods and linear independence over the field of algebraic numbers, we establish the existence of large families of such groups with rich algebraic and topological structure. We prove that there exist ( 2 c ) $(2<^>{\mathfrak {c}}) $ left parenthesis 2 Superscript German c Baseline right parenthesis pairwise distinct Liouville number groups generated by strong Liouville numbers and that, among these, there are ( 2 c ) $(2<^>{\mathfrak {c}})$ left parenthesis 2 Superscript German c Baseline right parenthesis pairwise nonhomeomorphic groups. We further show that there exist continuum many countable Liouville number groups, each generated by countably many strong Liouville numbers and homeomorphic, as topological spaces, to the rational numbers. In addition, we prove that no subgroup of the real numbers is homeomorphic to the space of Liouville numbers, thereby highlighting a strong topological distinction between the space of Liouville numbers and Liouville number groups.
Abstract Let $(\Omega , \mathfrak {M}, \mu )$ be a measure space. We show that if a surjective map $f: S_{L^1(\mu )} \to S_Y$ between the unit spheres of real $L^1(\mu )$ and of an arbitrary real normed space Y satisfies $$ \begin{align*} \min\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\min\{\|x+y\|, \|x-y\|\},\quad x,y\in S_{L^1(\mu)}, \end{align*} $$ then there exists a phase function $\varepsilon : S_{L^1(\mu )} \to \{-1, 1\}$ such that $\varepsilon \cdot f$ is a surjective isometry from $S_{L^1(\mu )}$ onto $S_Y$ , and furthermore, this isometry can be extended to a linear isometry on the whole space $L^1(\mu )$ .
Abstract Given a finite group G , we denote by ν ( G ) $\nu (G)$ nu left parenthesis upper G right parenthesis the probability that two randomly chosen elements of G generate a nilpotent subgroup. We prove that if ν ( G ) > 1 / 12 , $\nu (G)> {1}/{12},$ nu left parenthesis upper G right parenthesis greater than 1 divided by 12 comma then G is solvable.
Let ( A , G , alpha ) $(A,G,\alpha )$ left parenthesis upper A comma upper G comma alpha right parenthesis be a partial dynamical system and let A & rtimes; alpha , r G $A times _{\alpha ,r} G$ upper A right normal factor semidirect product Underscript alpha comma r Endscripts upper G denote the associated reduced partial crossed product. We introduce the Haagerup property for partial actions of discrete groups on C & lowast; $C<^>*$ upper C Superscript asterisk -algebras. We prove that the partial crossed product A & rtimes; alpha , r G $A times _{\alpha ,r} G$ upper A right normal factor semidirect product Underscript alpha comma r Endscripts upper G has the Haagerup property if and only if both A and the partial action alpha $\alpha $ alpha have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product, and that of the underlying C & lowast; $C<^>*$ upper C Superscript asterisk -algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits of C & lowast; $C<^>*$ upper C Superscript asterisk -algebras and apply this result to study the Haagerup property of inductive limits of partial crossed products.
Abstract An abelian monogenic polynomial f ( x ) ∈ Z [ x ] $f(x)\in {\mathbb Z}[x]$ f left parenthesis x right parenthesis element of double struck upper Z left bracket x right bracket is a monic polynomial of degree N ≥ 2 $N\ge 2$ upper N greater than or equals 2 that is irreducible over Q ${\mathbb Q}$ double struck upper Q , such that the Galois group of f ( x ) $f(x)$ f left parenthesis x right parenthesis over Q ${\mathbb Q}$ double struck upper Q is abelian, and { 1 , θ , θ 2 , … , θ N − 1 } $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ StartSet 1 comma theta comma theta squared comma ellipsis comma theta Superscript upper N minus 1 Baseline EndSet is a basis for the ring of integers of Q ( θ ) ${\mathbb Q}(\theta )$ double struck upper Q left parenthesis theta right parenthesis , where f ( θ ) = 0 $f(\theta )=0$ f left parenthesis theta right parenthesis equals 0 . We determine all abelian monogenic trinomials of the form x 2 n + a x n + b $x^{2n}+ax^{n}+b$ x Superscript 2 n Baseline plus a x Superscript n plus b , where n , a , b ∈ Z $n,a,b\in {\mathbb Z}$ n comma a comma b element of double struck upper Z with n ≥ 1 $n\ge 1$ n greater than or equals 1 and a b ≠ 0 $ab\ne 0$ a b not equals 0 .
Abstract We prove a new characterisation of slices embedded into some warped products in terms of higher-order mean curvatures.
Abstract Ballantine et al. [‘Partitions and elementary symmetric polynomials: an experimental approach’, Ramanujan J. 66 (2) (2025), Article no. 34] proposed two conjectures on the injectivity of a class of maps p r e k ${pre}_k$ p r e Subscript k defined on integer partitions. These maps arise from applying the sequence of elementary symmetric polynomials to integer partitions. We provide an infinite family of examples to disprove the conjecture for k ≥ 3 $k\ge 3$ k greater than or equals 3 and state a modified version of it. Throwing fresh light on this class of maps, we study the inter-relationships between them, deviating from the approaches so far, which study these maps one at a time. While the conjecture for k = 2 $k=2$ k equals 2 has now been settled, we provide alternate proofs of three subcases. We also discuss lower bounds for the number of partitions of n that are in the image of the map p r e 2 $pre_2$ p r e 2 .
Abstract For each pair j , n $j,n$ j comma n with n > j $n>j$ n greater than j , we consider the subgroup of the multiplicative group mod n of residues with order dividing n − j $n-j$ n minus j . We generalise some results in the case j = 1 $j=1$ j equals 1 due to Erdős and the current author [‘On the number of false witnesses for a composite number’, Math. Comp. 46 (1986), 259–279]. The case j = 0 $j=0$ j equals 0 is of particular interest.
For any integer k and any positive integer n , let $\sigma _k(n)=\sum _{d\mid n}d^k$ . For any prime p and any positive integer m , let $\nu _p(m)$ be the largest integer $\alpha $ such that $p^\alpha \mid m$ and let $\lceil x\rceil $ denote the least integer not less than x . In 2021, Amdeberhan et al. [‘Arithmetic properties of the sum of divisors’, J. Number Theory 223 (2021), 325–349] proved that $\nu _2(\sigma _1(n))\le \lceil \log _2n\rceil $ for any positive integer n and that $\nu _p(\sigma _1(n))\le \lceil \log _pn\rceil $ for any odd prime p if n satisfies some conditions. Recently, Zhao and Chen [‘ p -adic valuation of the sum of divisors’, Front. Math. 20 (4) (2025), 795–827] proved this unconditionally. We generalise these results to all k : for any prime p , any n and any $k\ge 2$ , $\nu _p(\sigma _k(n))\le \lceil k\log _p n\rceil .$ Let $p^\star $ be an odd prime. We also prove that there are an integer $k\ge 2$ and a prime q satisfying $\nu _{p^\star }(\sigma _k(q))=\lceil k\log _{p^\star }q\rceil $ if and only if $p^\star $ is a Fermat prime.
Let p and q be two primes with $(p,q)\equiv (1,5)$ or $(7,3) \pmod 8$ . Lagrange [‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16 (1) (1974–1975), Article no. 16] and Qin [‘Congruent numbers, quadratic forms and $K_2$ ’, Math. Ann. 383 (3–4) (2022), 1647–1686] showed that if $(\frac {q}{p})=-1$ , then $2pq$ is not a congruent number. By using Qin’s method, we prove that if $(p,q)\equiv (1,5) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-pq)\not \equiv p-1 \pmod {16}$ , then $2pq$ is not a congruent number; if $(p,q)\equiv (7,3) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-2pq)\not \equiv p+1 \pmod {16}$ , then $2pq$ is not a congruent number. Here, $h(-d)$ denotes the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-d})$ .
In 1985, Adams ['The algebraic independence of certain Liouville continued fractions', Proc. Amer. Math. Soc. 4 (1985), 512-516] proved that any finite number of Liouville series in different bases are algebraically independent. In this paper, we prove the equivalence between algebraic and linear independence when the bases are equal.
A positive integer N is said to be an infinitary harmonic number (IHN) if the harmonic mean of its infinitary divisors is integral. It is still an open problem whether or not there exist infinitely many IHNs. Cohen and Hagis ['Infinitary harmonic numbers', Bull. Aust. Math. Soc. 41 (1989), 151-158] showed that there exist at most finitely many IHNs with a fixed number of I-components. We give an explicit Nielsen-type upper bound for the number of IHNs. We also consider a class of general infinitary amicable numbers and obtain the Borho-type upper bound for them.
We prove a Helly-type theorem for dual p-operator spaces. As a consequence, any dual p-operator space is weakly p-locally reflexive.
