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Let p be a prime that divides the order of the group G. We show that a finite solvable group has class number at least f(p) where (f(p):=min {x+frac{p-1}{x}: xin mathbb {N}, x mid (p-1)}). We also obtain some applications to character degrees.

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely:

where (pin ]1,+infty [,) (Omega ) is a bounded, convex domain in ({mathbb {R}}^{N},) (nu _{Omega }) is its Euclidean outward normal, (beta ) is a real number, and F is a sufficiently smooth norm on ({mathbb {R}}^{N}.) We show an upper bound for (lambda _{F}(beta ,Omega )) in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on (beta ) and on the volume and the anisotropic perimeter of (Omega ,) in the spirit of the classical estimates of Pólya (J Indian Math Soc (NS) 24:413–419, 1961) for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity

when (beta >0.) The obtained results are new also in the case of the classical Euclidean Laplacian.

The purpose of this work is to study the conformal geometry of complete quasi Yamabe gradient solitons, which correspond to an interesting generalization for gradient Yamabe solitons. In this sense, we present a rigidity result for complete quasi Yamabe gradient solitons with constant scalar curvature. Moreover, we prove that quasi Yamabe gradient solitons can be conformally changed to constant scalar curvature.

We give a complete description of self-majorizing elements of Archimedean unital f-algebras. As an application, we furnish a new characterization of self-majorizing elements of Archimedean vector lattices.

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and (1 < n_1 le cdots le n_k), there exists an integer-valued polynomial on D, that is, an element of ({{,textrm{Int},}}(D) = { f in K[X] mid f(D) subseteq D }), which has precisely k essentially different factorizations into irreducible elements of ({{,textrm{Int},}}(D)) whose lengths are exactly (n_1, ldots , n_k). Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Galois compatible bijections between Brauer characters to virtual Morita equivalences. As a corollary, we obtain that the block version of Navarro’s refinement of Alperin’s weight conjecture holds for blocks with cyclic and Klein four defect groups, blocks of symmetric and alternating groups with abelian defect groups, and p-Blocks of (textrm{SL}_2(q)) and (textrm{GL}_2(q)), where p|q.

Let G be a finite group and p a prime dividing its order |G|, with p-part (|G|_p), and let (G_p) denote the set of all p-elements in G. A well known theorem of Frobenius tells us that (f_p(G)=|G_p|/|G|_p) is an integer. As (G_p) is the union of the Sylow p-subgroups of G, this Frobenius ratio (f_p(G)) evidently depends on the number (s_p(G)=|textrm{Syl}_p(G)|) of Sylow p-subgroups of G and on Sylow intersections. One knows that (s_p(G)=1+kp) and (f_p(G)=1+ell (p-1)) for nonnegative integers (k, ell ), and that (f_p(G)<s_p(G)) unless G has a normal Sylow p-subgroup. In order to get lower bounds for (f_p(G)) we, study the permutation character ({pi }={pi }_p(G)) of G in its transitive action on (textrm{Syl}_p(G)) via conjugation (Sylow character). We will get, in particular, that (f_p(G)ge s_p(G)/r_p(G)) where (r_p(G)) denotes the number of P-orbits on (textrm{Syl}_p(G)) for any fixed (Pin textrm{Syl}_p(G)). One can have (ell ge kge 1) only when P is irredundant for (G_p), that is, when P is not contained in the union of the (Qne P) in (textrm{Syl}_p(G)) and so (widehat{P}=bigcup _{Qne P}(Pcap Q)) a proper subset of P. We prove that (ell ge k) when (|widehat{P}|le |P|/p).

This paper presents a fairly general version of the well-known Gleason–Kahane–(dot{text {Z}})elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials (mathcal {P}). We do not make any topological assumptions on (mathcal {P}). We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces.

Ramanujan’s partition congruences modulo (ell in {5, 7, 11}) assert that

where (0<delta _{ell }<ell ) satisfies (24delta _{ell }equiv 1pmod {ell }.) By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes (ell ge 5,) we give a new proof of the conclusion that there are infinitely many m for which (p(ell m+delta _{ell })) is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies (m<(ell ^2-1)/24,) representing a significant improvement to the previous bound.

We generalize Gauss’ lemma over function fields, and establish a reciprocity law for power residue symbols. As an application, a reciprocity law for power residue symbols is established in totally imaginary function fields.