scholarly journals Diameters of random Cayley graphs of finite nilpotent groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniel El-Baz ◽  
Carlo Pagano
Keyword(s):  
Nilpotent Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Limiting Distribution ◽  
General Inequality ◽  
Bounded Rank

Abstract We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

Some properties of subgroup complementary addition Cayley graphs on abelian groups

2021 ◽  
Author(s):  
Naveen Palanivel ◽  
Chithra A. Velu
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Graph Invariants

In this paper, we introduce subgroup complementary addition Cayley graph [Formula: see text] and compute its graph invariants. Also, we prove that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text] where [Formula: see text].

scholarly journals On almost finitely generated nilpotent groups

1996 ◽  
Vol 19 (3) ◽  
pp. 539-544 ◽  
Author(s):  
Peter Hilton ◽  
Robert Militello
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Local Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Finitely Generated

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

scholarly journals Word problems for finite nilpotent groups

2020 ◽  
Vol 115 (6) ◽  
pp. 599-609
Author(s):  
Rachel D. Camina ◽  
Ainhoa Iñiguez ◽  
Anitha Thillaisundaram
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Character Degrees ◽  
Character Theory ◽  
Odd Order ◽  
Related Group

AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

scholarly journals Isomorphisms of Cayley digraphs of Abelian groups

1998 ◽  
Vol 57 (2) ◽  
pp. 181-188 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Prime Divisor ◽  
Open Problem ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Vertex Set ◽  
A Finite Group

For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

scholarly journals Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

10.37236/2369 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Integral Spectrum ◽  
Distance Matrices

It is shown that distance powers of an integral Cayley graph over an abelian group $\Gamma$ are again integral Cayley graphs over $\Gamma$. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

scholarly journals Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

2018 ◽  
Vol 49 (3) ◽  
pp. 183-194
Author(s):  
Ali Reza Ashrafi ◽  
Bijan Soleimani
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Prime Numbers ◽  
Graph Of Groups ◽  
Odd Order ◽  
Edge Transitive

Suppose $p$ and $q$ are odd prime numbers. In this paper, the connected Cayley graph of groups of order $3pq$, for primes $p$ and $q$, are investigated and all connected normal $\frac{1}{2}-$arc-transitive Cayley graphs of group of these orders will be classified.

scholarly journals Odd order nilpotent groups of class two with cyclic centre

1974 ◽  
Vol 17 (2) ◽  
pp. 142-153 ◽  
Author(s):  
Y. K. Leong
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Central Product ◽  
Odd Order ◽  
Image Position

The isomorphism problem for finite groups of odd order and nilpotency class 2 with cyclic centre will be solved using some results of Brady [1], [2]. Since a finite nilpotent group is the direct product of its Sylow subgroups, we only need to consider finite q-groups where q is a prime. It has been shown in [1] and [2] that a finite q-group of nilpotency class 2 with cyclic centre is a central product either of two-generator subgroups with cyclic centre or of two-generator subgroups with cyclic centre and a cyclic subgroup, and that the q-groups of class 2 on two generators with cyclic centre comprise the following list: , and if q = 2 we have as well .

scholarly journals Which Haar graphs are Cayley graphs?

10.37236/5240 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
István Estélyi ◽  
Tomaž Pisanski
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cayley Graph ◽  
Regular Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Equivalent Condition ◽  
Dihedral Groups ◽  
A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$. 

scholarly journals Improved Expansion of Random Cayley Graphs

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Po-Shen Loh ◽  
Leonard J. Schulman
Keyword(s):  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Expected Value ◽  
Small Eigenvalue ◽  
Irreducible Representations ◽  
Absolute Value ◽  
Graph Expansion ◽  
International Audience

International audience In Random Cayley Graphs and Expanders, N. Alon and Y. Roichman proved that for every ε > 0 there is a finite c(ε ) such that for any sufficiently large group G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of the Cayley graph with respect to c(ε ) log |G| random elements is less than ε . We reduce the number of elements to c(ε )log D(G) (for the same c), where D(G) is the sum of the dimensions of the irreducible representations of G. In sufficiently non-abelian families of groups (as measured by these dimensions), log D(G) is asymptotically (1/2)log|G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner84 and AlonMilman84-2,AlonMilman84-1. For any specified eigenvalue or expansion, therefore, random Cayley graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.

scholarly journals Integral Mixed Cayley Graphs over Abelian Groups

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Monu Kadyan ◽  
Bikash Bhattacharjya
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Mixed Graph ◽  
Vertex Set ◽  
Gamma S

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

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