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Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions H is the only finite-dimensional real division algebra. We first give a short elementary p...

We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $\varphi$ has infinite Reidemeister number $R(\varphi)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(\varphi)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[\rho]\mapsto[\rho\circ\varphi]}$., Comment: 9 pages. Author affiliations updated

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers R , the complex numbers C or the quaternions H ; (Complex version) Every complex normed division algebra is isometrically isomorphic to C . This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues.

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized $F$-signature of a ring of invariants by the second author and Nakajima to the modular case., 25 pages

For an arbitrary matrix A and the the Frobenius norm , it is proved thatIn addition, we derive inequalities for which are attained for normal matrices.

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A and the smash product A#H⁎ with respect to the Frobenius and symmetric properties.

In this paper, we give formulas for the embedding dimension, the Frobenius number, the type and the genus for a numerical semigroups generated by the Mersenne numbers greater than or equal to a given Mersenne number.

Let G be a finite group and e a positive integer dividing |G|, the order of G. The size of the set Le(G)={x∈G|xe=1} was studied originally by G. Frobenius, in order to find restrictions on the structure of G. Heineken and Russo [4] introduced B(G)=max{|Le(G)|e|e∈Div(exp(G))} as global breadth in the sense of Frobenius. In this paper, we investigate the groups G with B(G) = 4.

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in such a way that the structure is locally the span of $\frac{\partial}{\partial t_1},\ldots, \frac{\partial}{\partial t_r},\frac{\partial}{\partial \overline{z}_1},\ldots, \frac{\partial}{\partial \overline{z}_n}$; where $\mathbb{R}^r\times \mathbb{C}^n$ has coordinates $(t_1,\ldots, t_r, z_1,\ldots, z_n)$. In this paper, we give optimal regularity for the coordinate charts which achieve this realization. Namely, if the manifold has Zygmund regularity of order $s+2$ and the structure has Zygmund regularity of order $s+1$ (for some $s>0$), then the coordinate chart may be taken to have Zygmund regularity of order $s+2$. We do this by generalizing Malgrange's proof of the Newlander-Nirenberg Theorem to this setting., v3: 39 pages, final version, to appear in J. Funct. Anal

Let n be a positive integer and T ( n ) the numerical semigroup generated by { 3.2 n + i − 1 | i ∈ N } . In this paper, we give formulas for the Frobenius number, the gender, the embedding dimension and the type of T ( n ) .

We derive the lower bound for Frobenius number of symmetric (not complete intersection) semigroups generated by four elements., 3 pages

We derive some new conditions for integrability of dynamically defined C^1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C^1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability., Comment: 12 pages

We prove that the semigroup algebra of an ample semigroup \(S\) over a field is Frobenius if and only if \(S\) is a finite inverse semigroup.

A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.

We find the conditions under which the set-theoretic union of the kernels of two-generated Frobenius subgroups of a group G with fixed cyclic complement of order 3n is a normal subgroup in G.

Given a finite group G and an integer e ≥ 1 dividing the order of G , the size of the set L e ( G ) = { x ∈ G | x e = 1 } was studied originally by Frobenius, in order to find restrictions on the structure of G . The aim of the present paper is to classify groups by B ( G ) = max ⁡ { | L e ( G ) | e | e ∈ Div ( exp ⁡ ( G ) ) } , where Div ( exp ⁡ ( G ) ) is the set of all divisors of the exponent exp ⁡ ( G ) of G . We will show general statements regarding the center and the central quotient of G , by looking at B ( G ) . This improves some recent contributions of W. Meng and J. Shi in [7] .

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation $x^n=1$ in a group, the Solomon theorem (1969) on the number of solutions in a group to a system of equations having fewer equations than unknowns, and the Iwasaki theorem (1985) on roots of subgroups. There are other curious corollaries on groups and rings., Comment: 9 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: misprints corrected, an acknowledgement added. V3: misprints corrected

The classification of minimal algebraic surfaces in positive characteristics was accomplished by Bombieri and Mumford in the 1970s. In this work we decide completely which minimal algebraic surfaces in positive characteristics allow a lifting of their Frobenius over the truncated Witt rings of length 2. Besides, we show that the Frobenius morphism of many projective rational surfaces cannot be lifted to \(W_2(k)\).

Let n be any positive integer. We give a simple proof to the theorem that there always exist positive integers a, b and c such that the Frobenius number generated by a;b;c is equal to n.

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1]., 8 pages

We describe how Frobenius Weil algebras can be determined and underline the importance of Frobenius Weil bundles in several characteristic constructions in differential geometry.

In this paper, we present an efficient algorithm to compute the set of saturated numerical semigroups with a given Frobenius number.

We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra H. If H is a group Hopf algebra, we study a more general Frobenius type property, uncover the structure of graded Frobenius algebras, and investigate graded symmetric algebras. The graded Frobenius concept is related to Frobenius functors.

A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as G=K@?H with K a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K with connection set S an H-orbit on K generating K, where H is of even order or S consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel K is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its forwarding indices, Wiener indices and minimum gossip time. We also prove that its broadcasting time is equal to its diameter plus two or three. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein-Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein-Jacobi integers. We also prove that larger Eisenstein-Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for 6-valent first-kind Frobenius circulants. As a corollary any Eisenstein-Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein-Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. A distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.

In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field. The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method solves this dual problem in the F-finite case. © 2013 Elsevier B.V.

A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that F H/[F, F] is a Frobenius group with Frobenius kernel F/[F, F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms F H aiming at restrictions on G in terms of CG(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.

Connections between annihilators and ideals in Frobenius and symmetric algebras are used to provide a new proof of a result of Nakayama on quotient algebras, and an application is given to central symmetric algebras.

This paper deals with the orthogonal projection (in the Frobenius sense) A N of the identity matrix I onto the matrix subspace A S ( A ∈ R n × n , S being an arbitrary subspace of R n × n ). Lower and upper bounds on the normalized Frobenius condition number of matrix A N are given. Furthermore, for every matrix subspace S ⊂ R n × n , a new index κ F ( A , S ) , which generalizes the normalized Frobenius condition number of matrix A , is defined and analyzed.

In this article, we first investigate the properties of modular Frobenius groups. Then, we consider the case that G' is a minimal normal subgroup of a modular Frobenius group G. We give the complete classification of G when G' as a modular Frobenius kernel has no more than four conjugacy classes in G.

Self-dual codes constitute the most fascinating class of codes, by their many connections to invariant theory, combinatorial designs, and Euclidean lattices. In this short chapter, we focus our attention on existence conditions for alphabets having the structure of, successively, chain rings, commutative Frobenius rings, and noncommutative Frobenius rings. Necessary congruence existence conditions bearing on the size of the residue field are given. Sufficient existence conditions are derived in short lengths and extended to longer lengths by taking direct sums of codes.

In this paper we introduce the concept of integral Frobenius to formulate an integral analogue of the classical compatibility condition linking the collection of rational Tate modules V λ (A) arising from abelian varieties over number fields with real multiplication. Our main result gives a recipe for constructing an integral Frobenius when the real multiplication field has class number one. By exploiting algorithms already existing in the literature, we investigate this construction for three modular abelian surfaces over Q.

Ein Grundproblem der Algebra ist die Bestimmung der Galoisgruppe eines separablen Polynoms in einer Variablen. Liegen die Koeffizienten des Polynoms in einem endlichen Korper der Kardinalitat q, so ist diese Galoisgruppe erzeugt von dem Bild des FrobeniusAutomorphismus x 7→ x n . Hat das Polynom zusatzlich die spezielle Form a0X + a1X q + . . . + adX q mit a0, ad 6= 0, so wird die Operation von Frobenius durch eine Matrix in GLd(Fq) reprasentiert. Der vorliegende Artikel beantwortet die Frage, welche Matrizen auf diese Weise auftreten konnen fur gegebene q, n und d. In gewissem Sinn lost dies eine Variante des “Umkehrproblems der Galoistheorie” uber endlichen Korpern.

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law., 32 pages; submitted

In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ℤ2-graded algebras. Essentially, the results are the same as in the classical case, except that the notion of a ℤ2-graded division algebra needs to be modified. We classify all finite dimensional ℤ2-graded division algebras over ℂ and ℝ.

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $\phi$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $\phi$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2., Comment: 15 pages, many statements clarified and numerous other substantial improvements to the exposition (thanks to the referees). To appear in the Journal of Symbolic Computation

Using a geometric interpretation of continued fractions, we give a new proof of Rodseth’s formula for Frobenius numbers.

The purpose of this article is to discuss the conjecture on the Frobenius norm of matrices. The conjecture on the Frobenius norm of matrices proposed by Sloane and Harwit is proved for a class of special matrices.

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≅ ℤ2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups D* is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and is divisible.

Let $N \geq 2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this. We use techniques from the Geometry of Numbers to give upper and lower bounds on the $s$-Frobenius number for any nonnegative integer $s$., We include an appendix with an erratum and addendum to the published version of this paper: two inaccuracies in the statement of Theorem 2.2 are corrected and additional bounds on s-Frobenius numbers are derived

The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to C. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions., Comment: Some changes suggested by the referee, accepted for publication in Journal of Algebra