scholarly journals Some projective representations of finite abelian groups

1987 ◽  
Vol 29 (2) ◽  
pp. 197-203 ◽  
Author(s):  
A. O. Morris ◽  
M. Saeed-Ul-Islam ◽  
E. Thomas
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Projective Representations ◽  
Complete Sets ◽  
Image Position

In this paper, we continue the work initiated by Morris [5] and Saeed-ul-Islam [6,7] and determine complete sets of inequivalent irreducible projective representations (which we shall write as i.p.r.) of finite Abelian groups with respect to some additional factor sets.We consider an Abelian groupwhich will be referred to as an Abelian group of type (a1, …, am).

A Spectral Sequence for Cohomotopy

1969 ◽  
Vol 21 ◽  
pp. 712-729 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Abelian Group ◽  
Spectral Sequence ◽  
Prime Number ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Image Position

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.Theorem. Suppose that(i) Hi(X;Z) = 0 for i > k,(ii) for i > k – dThen there exists a spectral sequence withand the differential is given by

scholarly journals Representations of finite abelian groups Cnm,p

1985 ◽  
Vol 26 (2) ◽  
pp. 133-140 ◽  
Author(s):  
M. Saeed-ul-Islam
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Projective Representation ◽  
Complex Numbers ◽  
Finite Abelian Groups ◽  
Image Position ◽  
Factor Set

Let C= CmXCmX…XCm be the finite abelian group of order mn generated by n elements w1…,wn of order m. Let C be the field of complex numbers and P a projective representation of G with factor set α over C (see Morris [2]). Further letand

A First Approximation to {X, Y}

1969 ◽  
Vol 21 ◽  
pp. 702-711 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Homotopy Theory ◽  
Abelian Groups ◽  
Classification Theorem ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
First Approximation ◽  
Image Position

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.Our aim in this paper is to prove the following theorem.THEOREM. Iƒ ℭ is a class of finite abelian groups and(i) πi(Y) ∈ℭ for i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z)∈ ℭ for i > n + k,ThenThis statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

scholarly journals EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

2019 ◽  
Vol 150 (4) ◽  
pp. 1937-1964 ◽  
Author(s):  
Hua-Lin Huang ◽  
Zheyan Wan ◽  
Yu Ye
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Bar Resolution ◽  
Chain Map ◽  
A Chain

AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

scholarly journals The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

ON A PARTITION PROBLEM OF FINITE ABELIAN GROUPS

2015 ◽  
Vol 92 (1) ◽  
pp. 24-31
Author(s):  
ZHENHUA QU
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Partition Problem ◽  
Finite Abelian Groups ◽  
Disjoint Sets ◽  
Ordered Pairs

Let$G$be a finite abelian group and$A\subseteq G$. For$n\in G$, denote by$r_{A}(n)$the number of ordered pairs$(a_{1},a_{2})\in A^{2}$such that$a_{1}+a_{2}=n$. Among other things, we prove that for any odd number$t\geq 3$, it is not possible to partition$G$into$t$disjoint sets$A_{1},A_{2},\dots ,A_{t}$with$r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.

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