Cyclic Group Of Prime Order Research Articles

Generic initial ideals of modular polynomial invariants

We study the generic initial ideals (gin) of certain ideals that arise in modular invariant theory. For all cases an explicit generating set is known we compute the generic initial ideal of the Hilbert ideal of a cyclic group of prime order for all monomial orders. We also consider the Klein four group and note that its Hilbert ideals are Borel fixed with certain orderings of the variables. In all situations we consider, it is possible to select a monomial order such that the gin of the Hilbert ideal is equal to its initial ideal. Along the way we show that gin respects a permutation of the variables in the monomial order.

Quasimodular moonshine and arithmetic connections

We prove the existence of a module for the largest Mathieu group, whose trace functions are weight 2 2 quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain divisibility conditions on the number of F p \mathbb {F}_p points on Jacobians of modular curves. Extending such expressions to arbitrary primes, we find trace functions for modules of cyclic groups of prime order with similar connections. Moreover, for cyclic groups we give an explicit vertex operator algebra construction whose trace functions are given only in terms of weight 2 2 Eisenstein series.

Finite groups with a certain number of values of the Chermak–Delgado measure

In this paper, we study the finite groups whose Chermak–Delgado measure has exactly [Formula: see text] values. We will focus especially on the case [Formula: see text]. These groups determine an interesting class of [Formula: see text]-groups containing cyclic groups of prime order and extraspecial [Formula: see text]-groups.

A finiteness result for circulant core complex Hadamard matrices

We show that for every prime number p there exist finitely many circulant core complex Hadamard matrices of size p+1. The proof uses a ‘derivative at infinity’ argument to reduce the problem to Tao's uncertainty principle for cyclic groups of prime order (see [13]).

Dioid partitions of groups

A partition of a group is a dioid partition if the following three conditions are met: The setwise product of any two parts is a union of parts, there is a part that multiplies as an identity element, and the inverse of a part is a part. This kind of a group partition was first introduced by Tamaschke in 1968. We show that a dioid partition defines a dioid structure over the group, analogously to the way a Schur ring over a group is defined. After proving fundamental properties of dioid partitions, we focus on three part dioid partitions of cyclic groups of prime order. We provide classification results for their isomorphism types as well as for the partitions themselves.

Stanley–Reisner rings of simplicial complexes with a free action by an abelian group

We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley–Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzel’s calculation of the Hilbert series of some of the ring’s Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley and Adin that provide a lower bound on the h-vector of a Cohen–Macaulay complex admitting a free action by a cyclic group of prime order.

Noether’s problem for cyclic groups of prime order

Let k be a field and \(k(x_0,\ldots ,x_{p-1})\) be the rational function field of p variables over k where p is a prime number. Suppose that \(G=\langle \sigma \rangle \simeq C_p\) acts on \(k(x_0,\ldots ,x_{p-1})\) by k-automorphisms defined as \(\sigma :x_0\mapsto x_1\mapsto \cdots \mapsto x_{p-1}\mapsto x_0\). Denote by P the set of all prime numbers and define \(P_0=\{p\in P:\mathbb {Q}(\zeta _{p-1})\) is of class number one\(\}\) where \(\zeta _n\) a primitive n-th root of unity in \(\mathbb {C}\) for a positive integer n; \(P_0\) is a finite set by Masley and Montgomery (J Reine Angew Math 286/287:248–256, 1976). Theorem. Let k be an algebraic number field and \(P_k=\{p\in P: p\) is ramified in \(k\}\). Then \(k(x_0,\ldots ,x_{p-1})^G\) is not stably rational over k for all \(p\in P\backslash (P_0\cup P_k)\).

Symmetric powers and modular invariants of elementary abelian p-groups

Let E be an elementary abelian p-group of order q=pn. Let W be a faithful indecomposable representation of E with dimension 2 over a field k of characteristic p, and let V=Sm(W) with m<q. We prove that the rings of invariants k[V]E are generated by elements of degree ≤q and relative transfers. This extends recent work of Wehlau [18] on modular invariants of cyclic groups of order p. If m<p we prove that k[V]E is generated by invariants of degree ≤2q−3, extending a result of Fleischmann, Sezer, Shank and Woodcock [6] for cyclic groups of order p. Our methods are primarily representation-theoretic, and along the way we prove that for any d<q with d+m≥q, Sd(V⁎) is projective relative to the set of subgroups of E with order ≤m, and that the sequence Sd(V⁎)d≥0 is periodic with period q, modulo summands which are projective relative to the same set of subgroups. These results extend results of Almkvist and Fossum [1] on cyclic groups of prime order.

On equivariant and invariant topological complexity of smooth ℤ/_{𝕡}-spheres

We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear Z / p \mathbb {Z}/_{\!p} -spheres have both invariants either 2 2 or 3 3 and calculate exact values in all but two cases. On the other hand, we exhibit examples which show that these invariants can be arbitrarily large in the class of smooth Z / p \mathbb {Z}/_{\!p} -spheres.

A Note on Average of Roots of Unity

We consider the problem of characterizing all functions f defined on the set of integers modulo n with the property that an average of some nth roots of unity determined by f is always an algebraic integer. Examples of such functions with this property are linear functions. We show that, when n is a prime number, the converse also holds. That is, any function with this property is representable by a linear polynomial. Finally, we give an application of the main result to the problem of determining self-perfect isometries for the cyclic group of prime order p.

Invariable generation of iterated wreath products of cyclic groups

Given a sequence $$\{C_i\}_{i \in \mathbb N}$$ of cyclic groups of prime orders, let $$\Gamma _\infty $$ be the inverse limit of the iterated wreath products $$C_m \wr \cdots \wr C_2 \wr C_1$$ . We prove that the profinite group $$\Gamma _\infty $$ is not topologically finitely invariably generated.

SOLOMON’S ZETA FUNCTION OF Bp(Cp3 )

In 1977, Solomon L. introduced a zeta function for orders for which all ideals of finite index must be known. This work follows our previous research [Zeta functions of Burnside rings of groups of order p and p2, Comm. Algebra, 37(2009), 1758-1786], where we found the zeta function of the Burnside Ring for cyclic groups of prime order p and p2, respectively. The main objective of this paper is to obtain all ideals of finite index in Bp(Cp3 ) in order to determine ζBp(Cp3 ) (s) the zeta function of the Burnside Ring for a cyclic group of order p3..

Equivariant Alexander–Spanier cohomology and P.A. Smith theory

In Part 1 of the paper we give a new construction of equivariant Alexander–Spanier cohomology for actions of a finite group G, which is simpler and more direct than the original one given in [2]. In Part 2 we prove some useful results concerning equivariant Alexander–Spanier cohomology theory. In Part 3 we use equivariant Alexander–Spanier cohomology and the results proved in Part 2, in the case where G=Cp is a cyclic group of prime order p, to develop P.A. Smith theory from scratch, under minimal, in fact necessary, assumptions.

Planarity of the intersection graph of subgroups of a finite group

For a nontrivial finite group [Formula: see text] different from a cyclic group of prime order, the intersection graph [Formula: see text] of [Formula: see text] is the simple undirected graph whose vertices are the nontrivial proper subgroups of [Formula: see text] and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper we characterize all finite groups with planar intersection graphs. It turns out that few solvable groups have planar intersection graphs. Also we classify finite groups whose intersection graphs are bipartite, triangle free and forests.

An alternate approach to the Lie bracket on Hochschild cohomology

We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.

UNCERTAINTY PRINCIPLES FOR GROUPS AND RECONSTRUCTION OF SIGNALS

Uncertainty principles of harmonic analysis and their analogues for finite abelian groups are considered in the paper. Special attention is paid to the recent results of T. Tao and coauthors about cyclic groups of prime order. It is shown, that indicator functions of subgroups of finite Abelian groups are analogues of Gaussian functions. Finite-dimensional version of Poisson summation formula is proved. Opportunities of application of these results for reconstruction of discrete signals with incomplete number of coefficients are suggested. The principle of partial isometric whereby we can determine the minimum number of measurements for stable recovery of the signal are formulated.

A note on infinite aS-groups

Let G be a group. If every nontrivial subgroup of G has a proper supplement, then G is called an aS-group. We study some properties of aS-groups. For instance, it is shown that a nilpotent group G is an aS-group if and only if G is a subdirect product of cyclic groups of prime orders. We prove that if G is an aS-group which satisfies the descending chain condition on subgroups, then G is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an aS-group. Finally, it is shown that if G is an aS-group and |G| ≠ pq, p, where p and q are primes, then G has a triple factorization.

On Cohen–Macaulayness and depth of ideals in invariant rings

We investigate the presence of Cohen–Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a p-group is not Cohen–Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen–Macaulay factorial rings cannot contain Cohen–Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen–Macaulay, extending a result of Fleischmann.

On *-Clean Group Rings II

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3, and Q8. Most recently, Li, Yuan, and Parmenter gave a complete characterization of when the group algebra FCp is *-clean, where F is a field and Cp is a cyclic group of prime order p. In this article, we extend the above mentioned result from FCp to FqCpk, where Fq is a finite field and Cpk is a cyclic group of an odd prime power order pk. For the general case when G = Cn is cyclic group of order n, we also provide a necessary condition and a few sufficient conditions for FqCn to be *-clean.

On Noether’s problem for cyclic groups of prime order

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g\,|\,g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $k(G)=k(x_g\,|\,g\in G)^G$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_p$ of prime order $p$, it is unknown whether there exist infinitely many primes $p$ such that $\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$. Only known $17$ primes $p$ for which $\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$ are $p\leq 43$ and $p=61,67,71$. We show that for primes $p< 20000$, $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$ except for affirmative $17$ primes and undetermined $46$ primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$ for undetermined $28$ primes $p$ out of $46$.