Let n=p1n1p2n2⋯prnr, where r,n1,n2,…,nr are positive integers and p1,p2,…,pr are distinct prime numbers with p1<p2<…<pr. For the finite cyclic group Cn of order n, let P(Cn) be the power graph of Cn ...and κ(P(Cn)) be the vertex connectivity of P(Cn). It is known that κ(P(Cn))=p1n1−1 if r=1. For r≥2, we determine the exact value of κ(P(Cn)) when 2ϕ(p1p2⋯pr−1)≥p1p2⋯pr−1, and give an upper bound for κ(P(Cn)) when 2ϕ(p1p2⋯pr−1)<p1p2⋯pr−1, which is sharp for many values of n but equality need not hold always.
The problem of synchronization over a group G aims to estimate a collection of group elements G1⁎,…,Gn⁎∈G based on noisy observations of a subset of all pairwise ratios of the form Gi⁎Gj⁎−1. Such a ...problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup — an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.
In this paper we describe the automorphism groups of graphs and edge-colored graphs that are cyclic as permutation groups. In addition, we show that every such group is the automorphism group of a ...complete graph whose edges are colored with 3 colors, and we characterize those groups that are automorphism groups of simple graphs.
We compute the ring of non-induced representations for a cyclic group, C
n
, over an arbitrary field and show that it has rank
φ
(
n
)
, where
φ
is Euler's totient function-independent of the ...characteristic of the field. Along the way, we obtain a "pick-a-number" trick; expressing an integer n as a sum of products of p-adic digits of related integers.
Let Cn be a cyclic group of order n. A sequenceS of length ℓ over Cn is a sequence S=a1⋅a2⋅…⋅aℓ of ℓ elements in Cn, where a repetition of elements is allowed and their order is disregarded. We say ...that S is a zero-sum sequence if Σi=1ℓai=0 and that S is a zero-sum free sequence if S contains no zero-sum subsequence. In 2000, Gao obtained a construction of all zero-sum free sequences of length n−1−k over Cn for 0≤k≤n3.
In this paper, we consider a generalization for a random subset of Cn. Let R=R(Cn,p) be a random subset of Cn obtained by choosing each element in Cn independently with probability p. Let Nn−1−kR be the number of zero-sum free sequences of length n−1−k in R. Also, let Nn−1−k,dR be the number of zero-sum free sequences of length n−1−k having d distinct elements in R. We obtain the expectations of Nn−1−kR and Nn−1−k,dR for 0≤k≤n3 and show that Nn−1−kR and Nn−1−k,dR are asymptotically almost surely (a.a.s.) concentrated around their expectations when k is fixed. Moreover, we provide two ways to compute the expectations using partition numbers and a recurrence formula.
Given a subgroup H of a group (G,+), a (G,H,k,1) difference family (DF) is a set F of k-subsets of G such that {f−f′:f,f′∈F,f≠f′,F∈F}=G∖H. Let gZgh be the subgroup of order h in Zgh generated by g. A ...(Zgh,gZgh,k,1)-DF is called cyclic and written as a (gh,h,k,1)-CDF. This paper shows that for h∈{2,3,6}, there exists a (gh,h,4,1)-CDF if and only if gh≡h(mod12), g⩾4 and (g,h)∉{(9,3),(5,6)}. As a corollary, it is shown that a 1-rotational Steiner system S(2,4,v) exists if and only if v≡4(mod12) and v≠28. This solves the long-standing open problem on the existence of a 1-rotational S(2,4,v). As another corollary, we establish the existence of an optimal (v,4,1)-optical orthogonal code with ⌊(v−1)/12⌋ codewords for any positive integer v≡1,2,3,4,6(mod12) and v≠25. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic 3-ary constant-weight codes with weight four and minimum distance six.
A permutation group is a group of all permutations of some set. If the set that forms a permutation group is the n-first of natural number, then a permutation group is called a symmetry group. There ...is another type of group, i.e., a cyclic group and a dihedral group, and they are a subgroup of a symmetry group by numbering the vertices of the polygon. Sasirangan is the traditional batik from the South Kalimantan. There are 18 traditional patterns. All the patterns make some polygon. Because of this, the purpose of this research is to investigate the type of group that forms the patterns of Sasirangan. First, the authors give the procedure to investigate the patterns of Sasirangan, then use that procedure to the patterns of Sasirangan. The result of this research is the patterns of Sasirangan form cyclic groups C_1 and C_2, and dihedral groups D_2, D_4, D_5 and D_8.
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