Subgroup intersection graph of finite abelian groups

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

Let G be a finite abelian group with identity 0. For an integer the additive power graph of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some with When the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by More specifically, we obtain the diameter and the girth of the graph and its complement Using these, we obtain a condition for and its complement to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of and its complement and hence we prove that and its complement are weakly perfect. Also, we discuss about the perfectness of At last, we obtain a condition for and its complement to be vertex pancyclic.

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

If a finite group A of exponent 4 is the automorphism group of a torsionfree (abelian) group G, then A must be a subdirect product of cyclic groups C2 and C4 and of quaternion groups Qs (see [2, 4]); A is therefore contained in the variety V(Qs) generated by the quaternion group. Having dealt completely in [3] with the case of finite (or countable) abelian automorphism groups, we now turn to non-abelian groups (which, as we have mentioned before ([3], p. 34), must be of exponent 4 or 12). In this paper we confine ourselves to proving that for every n>2 the free n-generator group A, of V(Qs) is, in fact, the automorphism group of a suitably chosen torsionfree group 1. We also give at the end of the paper an example (with n = 3) of an epimorphic image of Aa of order 27 that cannot be an automorphism group, although it still satisfies the condition N3 of [2]: all its elements of order 2 lie in its centre. This group fails to satisfy the necessary condition N5 of [1] and cannot be a subdirect product of the kind described above.

Given a finite group G, the co-prime order graph of G is the simple undirected graph whose vertex set is G, and two distinct vertices x; y are adjacent if gcd(o(x); o(y)) = 1 or p, where p is a prime, and o(x) and o(y) are the orders of x and y, respectively. In this paper, we prove that, for a fixed positive integer k, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus k. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.

Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the graph-theoretic properties of the complement of the intersection graph of subgroups of G.

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.

Let [Formula: see text] be a non-trivial finite group different from a cyclic [Formula: see text]-group. The join graph of [Formula: see text] is a graph whose vertex set is the set of all proper subgroups of [Formula: see text] which are not contained in the Frattini subgroup of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are joined by an edge if and only if [Formula: see text]. In this paper we classify finite groups with regular graphs and determine their graphs.

In this paper, a new kind of graph on a finite group , namely the result involution graph is defined and studied. We use to denote this graph, is a simple undirected graph with vertex set. Two distinct vertices are adjacent if and only if their product is nontrivial involution element in . The result involution graph for several finite groups are obtained. We study some properties of the result involution graph by resizing graph by using the conjugacy classes of . Finally, we show that the result involution graphs for the symmetric groups and the alternating groups are connected with diameter at most 3 and radius at most 2 for. Furthermore, they have girth 3.

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

Let $G$ be a finite group. The reduced power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices $x$ and $y$ are adjacent if $\langle x \rangle \subset \langle y \rangle$ or $\langle y \rangle \subset \langle x \rangle$. In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a $\mathcal{P}$-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

In chemistry, the molecular structure can be represented as a graph. Based on the information from the graph, its characterization can be determined by computing the topological index. Topological index is a numerical value that can be computed by using some algorithms and properties of the graph. Meanwhile, the non-commuting graph is a graph, in which two distinct vertices are adjacent if and only if they do not commute, where it is made up of the non-central elements in a group as a vertex set. In this paper, the Szeged index of the non-commuting graph of some finite groups are computed. This paper focuses on three finite groups which are the quasidihedral groups, the dihedral groups, and the generalized quaternion groups. The construction of the graph is done by using Maple software. In finding the Szeged index, some of the previous results and properties of the graph for the quasidihedral groups, the dihedral groups, and the generalized quaternion groups are used. The generalisation of the Szeged index of the non-commuting graph is then established for the aforementioned groups. The results are then applied to find the Szeged index of the non-commuting graph of ammonia molecule.

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R ≅ Π 𝕑 2 n {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}} , n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.