Acta Mathematica Hungarica最新文献

A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. A subgroup A of a group G is called OS-propermutablein G if there is a subgroup B such that (G = NG(A)B), where AB is a subgroup of G and A permutes with all Schmidt subgroups of B. We proved (p)-solubility of a group in which a Sylow (p)-subgroup is OS-propermutable, where (pgeq 7) 7. For (p < 7) all non-Abelian composition factors of such group are listed.

In [18], we showed that the Boolean prime ideal theorem ((mathsf{BPI})) suffices to prove the celebrated theorem of R. Ellis, which states: ``Every compact Hausdorff right topological semigroup has an idempotent element''. However, the natural and intriguing question of the status of the reverse implication remained open until now. We resolve this open problem in the setting of (mathsf{ZFA}) (Zermelo–Fraenkel set theory with atoms), namely we establish that Ellis' theorem does not imply (mathsf{BPI}) in (mathsf{ZFA}), and thus is strictly weaker than (mathsf{BPI}) in (mathsf{ZFA}). From the above paper, we also answer two more open questions and strengthen some theorems.

Typical results are:

1. Ellis' theorem is true in the Basic Fraenkel Model, and thus Ellis' theorem does not imply (mathsf{BPI}) in (mathsf{ZFA}).

2. In (mathsf{ZF}) (Zermelo–Fraenkel set theory without the Axiom of Choice ((mathsf{AC}))), if (S) is a compact Hausdorff right topological semigroup with (S) well orderable, then every left ideal of (S) contains a minimal left ideal and a minimal idempotent element. In addition, every such semigroup (S) has a maximal idempotent element.

3. In (mathsf{ZF}), if (S) is a compact Hausdorff right topological abelian semigroup, then every left ideal of (S) contains a minimal left ideal.

4. In (mathsf{ZF}), (mathsf{BPI}) implies ``Every compact Hausdorff right topological abelian semigroup (S) has a minimal idempotent element''.

5. In (mathsf{ZFA}), the Axiom of Multiple Choice ((mathsf{MC})) implies ``Every compact Hausdorff right topological abelian semigroup (S) has a minimal idempotent element''.

6. In (mathsf{ZFA}), (mathsf{MC}) implies ``Every compact Hausdorff right topological semigroup (S) with (S) linearly orderable, has a minimal idempotent element''.

A convex geometry is a closure system satisfying the anti-exchange property. This paper, following the work of Adaricheva and Bolat [1] and the Polymath REU (2020), continues to investigate representations of convex geometries with small convex dimension by convex shapes on the plane and in spaces of higher dimension. In particular, we answer in the negative the question raised by Polymath REU (2020): whether every convex geometry of convex dimension 3 is representable by circles on the plane. We show there are geometries of convex dimension 3 that cannot be represented by spheres in any (mathbb{R}^k), and this connects to posets not representable by spheres from the paper of Felsner, Fishburn and Trotter [44]. On the positive side, we use the result of Kincses [55] to show that every finite poset is an ellipsoid order.

Many remarkable results have been obtained on important problems combining arithmetic properties of the integers and some restricted conditions of their digits in a given base. Maynard considered the number of the polynomial values with missing digits and gave an asymptotic formula. In this paper we study truncated polynomials with restricted digits by using the estimates for character sums and exponential sums modulo prime powers. In the case where the polynomials are monomial we further give exact identities.

We determine the solution of the Drygas functional equation that satisfies the additional condition ((y^2+y)f(x)= (x^2+x)f(y)) on a restricted domain. Also, some other properties of Drygas functions are given as well.

Let (K_{3}) be a non-normal cubic extension over (mathbb{Q}), and let (a_{K_{3}}(n)) be the (n)-th coefficient of the Dedekind zeta function (zeta_{K_{3}}(s)). In this paper, we investigate the asymptotic behaviour of the type

where (ellgeq 2) is any fixed integer. As an application, we also establish the asymptotic formulae of the variance of (a_{K_{3}}^{2}(n^{ell})). Furthermore, we also consider the asymptotic relations for shifted convolution sums associated to (a_{K_{3}}(n)) with classical divisor function.

Let (Gamma) be a finite connected pentavalent graph admitting a nonabelian simple arc-transitive automorphism group (T) and soluble vertex stabilizers. Let (p>|T|_{2}) be an odd prime and ((p,|T|)=1), where (|T|_{2}) is the largest power of 2 dividing the order (|T|) of (|T|). Then we prove that there exists a (p)-radical cover (widetilde{Gamma}) of (Gamma) such that the full automorphism group (text{Aut}(widetilde{Gamma})) of (widetilde{Gamma}) is equal to (O_{p}(text{Aut}(widetilde{Gamma})).T) and the covering transformation group is (O_{p}(text{Aut}(widetilde{Gamma}))), where (O_{p}(text{Aut}(widetilde{Gamma}))) is the (p)-radical of (text{Aut}(widetilde{Gamma})).

A bounded linear operator (T) on a Hilbert space (H) is said to be absolutely norm attaining ((T in mathcal{AN}(H))) if the restriction of (T) to any non-zero closed subspace attains its norm and absolutely minimum attaining ((T in mathcal{AM}(H))) if every restriction to a non-zero closed subspace attains its minimum modulus.

In this article, we characterize normal operators in (overline{mathcal{AN}(H)}), the operator norm closure of (mathcal{AN}(H)), in terms of the essential spectrum. Later, we study representations of quasinormal and hyponormal operators in (overline{mathcal{AN}(H)}). Explicitly, we prove that any hyponormal operator in (overline{mathcal{AN}(H)}) is a direct sum of a normal (mathcal{AN})-operator and a (2times2) upper triangular (mathcal{AM})-operator matrix. Finally, we deduce some sufficient conditions implying the normality of them.

We construct the lower radical class and the semisimple closurefor a given class using class operators and detail some of the properties of theseoperators and their interplay with the operators already used in radical theory.The setting is the class of algebras introduced by Puczy lowski which ensures theresults hold in groups, multi-operator groups such as rings, as well as loops andhoops.

Let (sigma ={sigma_{i} mid iin I}) be some partition of the set of all primes and (G) a finite group. Then (G) is said to be (sigma)-full if (G) has a Hall (sigma _{i})-subgroup for all (i); (sigma)-primary if (G) is a (sigma _{i})-group for some (i); (sigma)-soluble if every chief factor of (G) is (sigma)-primary; (sigma)-nilpotent if (G) is the direct product of (sigma)-primary groups; (G^{mathfrak{N}_{sigma}}) denotes the (sigma)-nilpotent residual of (G), that is, the intersection of all normal subgroups (N) of (G) with (sigma)-nilpotent quotient (G/N).

A subgroup (A) of (G) is said to be: (sigma)-permutable in (G) provided (G) is (sigma)-full and (A) permutes with all Hall (sigma _{i})-subgroups (H) of (G) (that is, (AH=HA)) for all (i); (sigma)-subnormal in (G) if there is a subgroup chain (A=A_{0} leq A_{1} leq cdots leq A_{n}=G) such that either (A_{i-1} trianglelefteq A_{i}) or (A_{i}/(A_{i-1})_{A_{i}}) is (sigma)-primary for all (i=1, ldots , n).

Let (A_{sigma G}) be the subgroup of (A) generated by all (sigma)-permutable subgroups of (G) contained in (A) and (A^{sigma G}) be the intersection of all (sigma)-permutable subgroups of (G) containing (A).

We prove that if (G) is a finite (sigma)-soluble group, then the (sigma)-permutability is a transitive relation in (G) if and only if (G^{mathfrak{N}_{sigma}}) avoids the pair ((A^{sigma G}, A_{sigma G})), that is, (G^{mathfrak{N}_{sigma}}cap A^{sigma G}= G^{mathfrak{N}_{sigma}}cap A_{sigma G}) for every (sigma)-subnormal subgroup (A) of (G).