Search Results (43)

In this paper, we investigate Noether’s problem concerning the rationality of the field of invariants $k\left(G\right)$ for finite abelian groups G of odd order. We establish necessary and sufficient conditions for the rationality of the extension $k\left(G\right)$ over an arbitrary field k of characteristic 0, providing a more flexible alternative to the classical requirement that all cyclic primary components be rational. Specifically, we present a criterion for elementary abelian q-groups $C_q^n$ in terms of the properties of certain norm maps (Theorem 4) and generalize these results to provide necessary and sufficient conditions for arbitrary abelian groups of odd order (Theorem 5). Furthermore, we provide a computational implementation of these criteria in PARI/GP and offer a concrete arithmetic classification of rational invariant fields over $\mathbb{Q}$ for groups of various odd orders. Full article

Characterizing specific types of units in group rings in terms of the Jacobson radical constitutes a frequently investigated problem in the theory of group rings. In this study, since R is a commutative ring with unity and G is a finite Abelian group, $U_{ft}\left(RG\right)$ as the set of feckly trivial units in the group ring $RG$, consisting of units that are congruent modulo $J\left(RG\right)$ to some $g\in G$, where $J\left(RG\right)$ denotes the Jacobson radical of $RG$, is defined. Secondly, some necessary and sufficient conditions are given for the group $U\left(RG\right)$ of units of the group ring $RG$ to be feckly trivial under the assumption that $supp\left(G\right)\cap jp\left(R\right)=\varnothing$ where $\mathcal{P}$ is the set of all prime integers, $supp\left(G\right)=\{p\in \mathcal{P}:G_p\ne e_G\}$, $jp\left(R\right)=\{p\in \mathcal{P}:\exists r\in R\setminus \left\{0_R\right\},pr\in J\left(R\right)\}$ and $G_p$ is the p-primary component in G. Finally, two open problems related to this notion are introduced. Full article

This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of $\mathbb{Q}$ and describe algebraic number fields through their realization as finite-dimensional $\mathbb{Q}$-algebras (via multiplication operators and matrix representations), leading naturally to the arithmetic invariants—trace, norm, and discriminant—and to the ring of integers, ideals, Dedekind domains, and the ideal class group. We then develop the classical theory of cyclotomic fields, emphasizing their Galois structure and their role in abelian extensions of $\mathbb{Q}$. Next, we discuss ramification in general extensions, including decomposition and inertia groups, the Frobenius element, and the Chebotarev density theorem. The exposition continues with a concise algebraic introduction to elliptic curves and their L-functions, and it places key conjectural links (including Birch and Swinnerton-Dyer) in context. Finally, a collection of examples highlights a common operational language between fractional calculus and number theory: Laplace and Mellin transforms turn convolution-type operators into multiplication, clarifying the appearance of $\Gamma$-factors, Dirichlet series, and zeta- and L-function structures in both settings. Full article

Let $A_{ex}\left(\mathcal{G}\right)$ be the extended adjacency matrix of G. The eigenvalues of $A_{ex}\left(\mathcal{G}\right)$ are called extended adjacency eigenvalues of $\mathcal{G}$. The sum of the absolute values of eigenvalues of the $A_{ex}$-matrix is called the extended adjacency energy ${\mathcal{E}}_{ex}\left(\mathcal{G}\right)$ of $\mathcal{G}$. In this paper, we obtain the $A_{ex}$-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the $A_{ex}$-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the $A_{ex}$-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the $A_{ex}$-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the $A_{ex}$-eigenvalues of the power graph of finite abelian groups. Full article

Here all rings have identities. Let R be a ring and let R-mod denote the additive category of left finitely generated R-modules. Note that if R is a noetherian ring, then R-mod is an abelian category and every R-module is a finite direct sum of indecomposable R-modules. Finite Group Modular Representation Theory concerns the study of left finitely generated $\mathcal{O}G$-modules where G is a finite group and $\mathcal{O}$ is a complete discrete valuation ring with $\mathcal{O}/J\left(\mathcal{O}\right)$ a field of prime characteristic p. Thus $\mathcal{O}G$ is a noetherian $\mathcal{O}$-algebra. The Green Theory in this area yields for each isomorphism type of finitely generated indecomposable (and hence for each isomorphism type of finitely generated simple $\mathcal{O}G$-module) a theory of vertices and sources invariants. The vertices are derived from the set of p-subgroups of G. As suggested by the above, in Basic Definition and Main Results for Rings Section, let $\mathrm{\Sigma }$ be a fixed subset of subrings of the ring R and we develop a theory of $\mathrm{\Sigma }$-vertices and sources for finitely generated R-modules. We conclude Basic Definition and Main Results for Rings Section with examples and show that our results are compatible with a ring isomorphic to R. For Idempotent Morita Equivalence and Virtual Vertex-Source Pairs of Modules of a Ring Section, let e be an idempotent of R such that $R=ReR$. Set $B=eRe$ so that B is a subring of R with identity e. Then, the functions $eR{\otimes }_R\ast :R-\mathrm{mod}\to B-\mathrm{mod}$ and $Re{\otimes }_B\ast :B-\mathrm{mod}\to R-\mathrm{mod}$ form a Morita Categorical Equivalence. We show, in this Section, that such a categorical equivalence is compatible with our vertex-source theory. In Two Applications with Idemptent Morita Equivalence Section, we show such compatibility for source algebras in Finite Group Block Theory and for naturally Morita Equivalent Algebras. Full article

For an undirected graph G, a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum $\mathbb{A}$-flows, which gives edge weights using nonzero elements of an additive Abelian group $\mathbb{A}$ and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: $\mathbb{A}={\mathbb{Z}}_k$, the finite cyclic group of integer congruence modulo k, and $\mathbb{A}=\mathbb{Z}$, the infinite cyclic group of integers. The constant sum under a constant-sum $\mathbb{A}$-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by $I_k\left(G\right)$ and $I\left(G\right)$ for $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$, respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases $\mathbb{Z}$ and ${\mathbb{Z}}_k$ have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs $W_n$ and fan graphs $F_n$ in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning ${\mathbb{Z}}_k$ and $\mathbb{Z}$, respectively. Some open problems will be mentioned in the concluding remarks. Full article

Hopf–Galois extensions extend the idea of principal bundles to noncommutative geometry, using Hopf algebras as symmetries. We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf–Galois extensions (quantum principal bundles) for certain finite abelian groups. The corresponding strong universal connections are computed. We show that $M_n\left(\mathbb{C}\right)$ is a trivial quantum principle bundle for the Hopf algebra $\mathbb{C}[{\mathbb{Z}}_n\times {\mathbb{Z}}_n]$. We conclude with an application relating calculi on groups to calculi on matrices. Full article

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from ${\mathbb{Z}}^E$ orthogonal to rows of the matrix form a regular chain group N. Assume that $\psi$ is a homomorphism from N into a finite additive Abelian group A and let $A_{\psi }\left[N\right]$ be the set of vectors g from ${(A-0)}^E$, such that ${\sum }_{e\in E}g\left(e\right)·f\left(e\right)=\psi \left(f\right)$ for each $f\in N$ (where · is a scalar multiplication). We show that $|A_{\psi }\left[N\right]|$ can be evaluated by a polynomial function of $\left|A\right|$. In particular, if $\psi \left(f\right)=0$ for each $f\in N$, then the corresponding assigning polynomial is the classical characteristic polynomial of M. Full article

Let ${\left(k_{\mu }\right)}_{\mu =1}^4$ be a quartet of cyclic cubic number fields sharing a common conductor $c=pqr$ divisible by exactly three prime(power)s, $p,q,r$. For those components of the quartet whose 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic, the automorphism group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ of the maximal metabelian unramified 3-extension of $k_{\mu }$ is determined by conditions for cubic residue symbols between $p,q,r$ and for ambiguous principal ideals in subfields of the common absolute 3-genus field $k^{*}$ of all $k_{\mu }$. With the aid of the relation rank $d_2\left(\mathfrak{M}\right)$, it is decided whether $\mathfrak{M}$ coincides with the Galois group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })$ of the maximal unramified pro-3-extension of $k_{\mu }$. Full article

This paper contributes to the classification of flag-transitive symmetric 2-$(v,k,\lambda )$ designs with $\lambda$ prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is $PS{L}_n\left(q\right)$. Full article

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups. Full article

Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group $\Gamma$. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms. Full article

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set $\mathcal{G}\left(Q\right)$ if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations $Q_p$ and $R_p$ are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback $H_t$ from the l-equivalences $H_i\to H$ and $H_j\to H$, $t\equiv (i+j)mods$, where $s=\left|\mathcal{G}\left(H\right)\right|$, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product $G\times K$ of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial. Full article

The factorization of groups into a Zappa–Szép product, or more generally into a k-fold Zappa–Szép product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group and has recently been applied to public-key cryptography. We provide a generalization of the k-fold Zappa–Szép product of cyclic groups, which we call $OGS$ decomposition. It is easy to see that the existence of an $OGS$ decomposition for all the composition factors of a non-abelian group G implies the existence of an $OGS$ for G itself. Since the composition factors of a soluble group are cyclic groups, it has an $OGS$ decomposition. Therefore, the question of the existence of an $OGS$ decomposition is interesting for non-soluble groups. The Jordan–Hölder theorem motivates us to consider the existence of an $OGS$ decomposition for finite simple groups. In 1993, Holt and Rowley showed that $PS{L}_2\left(q\right)$ and $PS{L}_3\left(q\right)$ can be expressed as a product of cyclic groups. In this paper, we consider an $OGS$ decomposition of $PS{L}_2\left(q\right)$ from a different point of view to that of Holt and Rowley. We look at its connection to the $BN$-$pair$ decomposition of the group. This connection leads to sequences over ${\mathbb{F}}_q$, which can be defined recursively, with very interesting properties, and are closely connected to Dickson and Chebyshev polynomials. Since every finite simple Lie-type group exhibits $BN$-$pair$ decomposition, the ideas in this paper might be generalized to further simple Lie-type groups. Full article

Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators. Full article