Cyclic Group Of Order Research Articles

On the classification of fake lens spaces

In the first part of the paper we present a classification of fake lens spaces of dimension ≥ 5 whose fundamental group is the cyclic group of order any N ≥ 2. The classification is stated in terms of the simple structure sets in the sense of surgery theory. The results use and extend the results of Wall and others in the cases N = 2 and N odd and the results of the authors of the present paper in the case N = 2K . In the second part we study the suspension map between the simple structure sets of lens spaces of different dimensions. As an application we obtain an inductive geometric description of the torsion invariants of fake lens spaces.

2-Graded polynomial identities for the Jordan algebra of the symmetric matrices of order two

The Jordan algebra of the symmetric matrices of order two over a field K has two natural gradings by Z2, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a Z2-grading on Bn, the Jordan algebra of a symmetric bilinear form in a vector space of dimension n (n=1,2,…,∞). We call this grading the scalar one since its even part consists only of the scalars. As a by-product we obtain finite bases of the Z2-graded identities for Bn. In fact the last result describes the weak Jordan polynomial identities for the pair (Bn,Vn).

The group of symmetries of the shorter Moonshine module

It is shown that the automorphism group of the shorter Moonshine module VB ♮ constructed in the author’s Ph.D. thesis (Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Math. Schr. 286: 1996) is the direct product of the finite simple group known as the Baby Monster and the cyclic group of order 2.

The symmetries of the fine gradings of sl(nk,C) associated with direct product of Pauli groups

A grading of a Lie algebra is called fine if it could not be further refined. For a fine grading of a simple Lie algebra, we define its Weyl group to describe the symmetry of this grading. It is already known that the Weyl group of the fine grading of sl(n,C) induced by the action of the group Πn of the generalized Pauli matrices of rank n is SL(2,Zn), where Zn is the cyclic group of order n. In this paper, we consider the fine grading of sl(nk,C) induced by the action of the group of k-fold tensor product of the generalized Pauli matrices of rank n. We prove that its Weyl group is Sp(2k,Zn) and is generated by transvections; therefore, this generalizes the previous result.

Formula omitted]-filtered modules and nilpotent orbits of a parabolic subgroup in [formula omitted

We study certain Δ -filtered modules for the Auslander algebra of k [ T ] / T n ⋊ C 2 where C 2 is the cyclic group of order two. The motivation of this lies in the problem of describing the P -orbit structure for the action of a parabolic subgroup P of an orthogonal group. For any parabolic subgroup of an orthogonal group we construct a map from parabolic orbits to Δ -filtered modules and show that in the case of the Richardson orbit, the resulting module has no self-extensions.

A Carlitz identity for the wreath product [formula omitted

We present in this work a new flag major index fmaj r for the wreath product G r , n = C r ≀ S n , where C r is the cyclic group of order r and S n is the symmetric group on n letters. We prove that fmaj r is equidistributed with the length function on G r , n and that the generating function of the pair ( des r , fmaj r ) over G r , n , where des r is the usual descent number on G r , n , satisfies a “natural” Carlitz identity, thus unifying and generalizing earlier results due to Carlitz (in the type A case), and Chow and Gessel (in the type B case). A q-Worpitzky identity, a convolution-type recurrence and a q-Frobenius formula are also presented, with combinatorial interpretation given to the expansion coefficients of the latter formula.

Symmetric Bowtie Decompositions of the Complete Graph

Given a bowtie decomposition of the complete graph $K_v$ admitting an automorphism group $G$ acting transitively on the vertices of the graph, we give necessary conditions involving the rank of the group and the cycle types of the permutations in $G$. These conditions yield non–existence results for instance when $G$ is the dihedral group of order $2v$, with $v\equiv 1, 9\pmod{12}$, or a group acting transitively on the vertices of $K_9$ and $K_{21}$. Furthermore, we have non–existence for $K_{13}$ when the group $G$ is different from the cyclic group of order $13$ or for $K_{25}$ when the group $G$ is not an abelian group of order $25$. Bowtie decompositions admitting an automorphism group whose action on vertices is sharply transitive, primitive or $1$–rotational, respectively, are also studied. It is shown that if the action of $G$ on the vertices of $K_v$ is sharply transitive, then the existence of a $G$–invariant bowtie decomposition is excluded when $v\equiv 9\pmod{12}$ and is equivalent to the existence of a $G$–invariant Steiner triple system of order $v$. We are always able to exclude existence if the action of $G$ on the vertices of $K_v$ is assumed to be $1$–rotational. If, instead, $G$ is assumed to act primitively then existence can be excluded when $v$ is a prime power satisfying some additional arithmetic constraint.

Linear and sublinear time algorithms for the basis of abelian groups

It is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G ≅ G 1 × G 2 × ⋯ × G t , where each G i is a cyclic group of order p j for some prime p and integer j ≥ 1 . If a i generates the cyclic group of G i , i = 1 , 2 , … , t , then the elements a 1 , a 2 , … , a t are called a basis of G . We show a randomized algorithm such that given a set of generators M = { x 1 , … , x k } for an abelian group G and the prime factorization of order ord ( x i ) ( i = 1 , … , k ) , it computes a basis of G in O ( | M | ( log n ) 2 + ∑ i = 1 t n i p i n i / 2 ) time, where n = | G | has prime factorization p 1 n 1 p 2 n 2 ⋯ p t n t (which is not a part of input). This generalizes Buchmann and Schmidt’s algorithm that takes O ( | M | | G | ) time. In another model, all elements in an abelian group are put into a list as a part of input. We obtain an O ( n ) time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group.

Morphic Properties of Extensions of Rings

An element a in a ring R is called left morphic if R/Ra ≅ ℓR(a), where ℓR(a) denotes the left annihilator of a in R. A ring R is said to be left morphic if every element is left morphic. In this paper, it is shown that if I is an ideal of a unit regular ring R, then for each positive integer n, [Formula: see text] is a left morphic ring. This extends two recent results of Lee and Zhou. It is also proved that if R is a strongly regular ring and Cn= 〈g〉 is a cyclic group of order n ≥ 2, then for any r ∈ R, 1 + rg is morphic in the group ring RCn.

ROBINSON-SCHENSTED CORRESPONDENCE FOR THE G-VERTEX COLORED PARTITION ALGEBRA

In this paper, we develop the Robinson-Schensted correspondence for the G-vertex colored partition algebras, which gives the bijection between the set of G-vertex colored partition diagrams Pk(n, G) and the pairs of [Formula: see text]-vacillating tableaux of shape λ, [Formula: see text], [Formula: see text] where Γk= {µ ⊢ m|0 ≤ m ≤ k}, Ψq= {λ | λ = (q - j, 1j), j = 0, 1,…, q - 1} and G is a cyclic group of order q. We also derive the Knuth relations for the G-vertex colored partition algebra by using the Robinson-Schensted correspondence for the [Formula: see text]-vacillating tableau of shape [Formula: see text].

On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.

2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.

Groups of Cube-Free Odd Order

(2010). Groups of Cube-Free Odd Order. The American Mathematical Monthly: Vol. 117, No. 4, pp. 363-365.

C*-Crossed-Products by an Order-Two Automorphism

AbstractWe describe the representation theory of C*-crossed-products of a unital C*-algebra A by the cyclic group of order 2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of A. We characterize each class in term of the restriction of the representations to the fixed point C*-subalgebra of A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.

Regular embeddings of [formula omitted] where [formula omitted] is a power of 2. II: The non-metacyclic case

The aim of this paper is to complete a classification of regular orientable embeddings of complete bipartite graphs K n , n , where n = 2 e . The method involves groups G which factorise as a product G = X Y of two cyclic groups of order n such that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G in the case where G is not metacyclic. We prove that for each n = 2 e , e ≥ 3 , there are up to map isomorphism exactly four regular embeddings of K n , n such that the automorphism group G preserving the surface orientation and the bi-partition of vertices is a non-metacyclic group, and that there is one such embedding when n = 4 .

A p-nilpotency criterion

Let G be a finite group, p a fixed prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, then G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p = 2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.

The Fujita combinatorial enumeration for the non-rigid group of 2,4-dimethylbenzene

Using non-rigid group theory, it was previously shown that the full non-rigid group of 2,4-dimethylbenzene is an ummatured and isomorphic to the group C2?(C3wrC2) of order 36, where Cn is the cyclic group of order n, the symbols ? and wr stand for the direct and wreath products, respectively. Herein, it is first shown that this group has 12 dominant classes. Then, the Markaracter Table, the Table of all integer-valued characters and the unit subduced cycle index (USCI) Table of the full non-rigid group of 2,4-dimethylbenzene are successfully derived for the first time.

Classes réalisables d'extensions métacycliques de degré lm

Let k be a number field and O k its ring of integers. Let l be a prime number and m a natural number. Let C (resp. H) be a cyclic group of order l (resp. m). Let Γ = C ⋊ H be a metacyclic group of order lm, with H acting faithfully on C. Let M be a maximal O k -order in the semi-simple algebra k [ Γ ] containing O k [ Γ ] , and Cl ( M ) its locally free class group. We define the set R ( M ) of realizable classes to be the set of classes c ∈ Cl ( M ) such that there exists a Galois extension N / k which is tame, with Galois group isomorphic to Γ, and for which [ M ⊗ O k [ Γ ] O N ] = c , where O N is the ring of integers of N. In the present article, we define a subset of R ( M ) and prove, by means of a description using a Stickelberger ideal, that it is a subgroup of Cl ( M ) , under the hypothesis that k and the l-th cyclotomic field over Q are linearly disjoint.

Indexes of unsplittable minimal zero-sum sequences of length [formula omitted

Let G be a finite abelian group and S = g 1 ⋯ g l a minimal zero-sum sequence of elements in G . We say that S is unsplittable if there do not exist an element g i ∈ supp ( S ) and two elements x , y ∈ G such that x + y = g i and S a − 1 x y is a minimal zero-sum sequence as well. The notion of the index of a minimal zero-sum sequence in G has been recently addressed in the mathematical literature (see Definition 1.1). Let I ( C n ) be the minimal integer t such that every minimal zero-sum sequence of at least t elements in C n (the cyclic group of order n ) satisfies index ( S ) = 1 . In this paper, all the unsplittable minimal zero-sum sequences of length I ( C n ) − 1 are discovered and their indexes are computed. The results show that Conjecture 1.1 is true when n is odd, and false when n is even.

Hopf Algebra Extensions of Group Algebras and Tambara-Yamagami Categories

We determine the structure of Hopf algebras that admit an extension of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension < 36 is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.