479 research outputs found

    On divisibility graph for simple Zassenhaus groups

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    The divisibility graph D(G)D(G) for a finite group GG is a graph with vertex set cs (G){1}cs~(G)\setminus\{1\} where cs (G)cs~(G) is the set of conjugacy class sizes of GG. Two vertices aa and bb are adjacent whenever aa divides bb or bb divides aa. In this paper we will find D(G)D(G) where GG is a simple Zassenhaus group

    Divisibility graph for symmetric and alternating groups

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    Let XX be a non-empty set of positive integers and X=X{1}X^*=X\setminus \{1\}. The divisibility graph D(X)D(X) has XX^* as the vertex set and there is an edge connecting aa and bb with a,bXa, b\in X^* whenever aa divides bb or bb divides aa. Let X=cs GX=cs~{G} be the set of conjugacy class sizes of a group GG. In this case, we denote D(cs G)D(cs~{G}) by D(G)D(G). In this paper we will find the number of connected components of D(G)D(G) where GG is the symmetric group SnS_n or is the alternating group AnA_n

    Quotient graphs for power graphs

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    In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph P0(G)\mathcal{P}_0(G) of a finite group GG, finding a formula for the number c(P0(G))c(\mathcal{P}_0(G)) of its components which is particularly illuminative when GSnG\leq S_n is a fusion controlled permutation group. We make use of the proper quotient power graph P~0(G)\widetilde{\mathcal{P}}_0(G), the proper order graph O0(G)\mathcal{O}_0(G) and the proper type graph T0(G)\mathcal{T}_0(G). We show that all those graphs are quotient of P0(G)\mathcal{P}_0(G) and demonstrate a strong link between them dealing with G=SnG=S_n. We find simultaneously c(P0(Sn))c(\mathcal{P}_0(S_n)) as well as the number of components of P~0(Sn)\widetilde{\mathcal{P}}_0(S_n), O0(Sn)\mathcal{O}_0(S_n) and T0(Sn)\mathcal{T}_0(S_n)

    The Divisibility Graph of finite groups of Lie Type

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    The Divisibility Graph of a finite group GG has vertex set the set of conjugacy class lengths of non-central elements in GG and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic

    Domination parameters and diameter of Abelian Cayley graphs

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    Using the domination parameters of Cayley graphs constructed out of Zp×Zm\mathbb{Z}_{p}\times \mathbb{Z}_{m}, where m{pα,pαqβ,pαqβrγ},m\in\{p^{\alpha}, p^{\alpha}q^{\beta}, p^{\alpha}q^{\beta}r^{\gamma}\}, in this paper we are discussing about the total and connected domination number and diameter of these Cayley graphs

    A New Characterization of PSL(2, q) for Some q

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    Let G be a finite group and let π e (G) be the set of orders of elements from G. Let k ∈ π e (G) and let m k be the number of elements of order k in G. We set nse (G) := {m k | k ∈ π e (G)}. It is proved that PSL(2, q) are uniquely determined by nse (PSL(2, q)), where q ∈ {5, 7, 8, 9, 11, 13}. As the main result of the paper, we prove that if G is a group such that nse (G) = nse (PSL(2, q)), where q ∈ {16, 17, 19, 23}, then G ≅ PSL(2, q).Нехай G — скінченна група, а πe(G) — множина порядків елемента з G. Нехай також k∈πe(G), а mk — число елементів порядку k в G. Покладемо nse (G):={mk|k∈πe(G)}. Доведено, що PSL(2,q) однозначно визначаються nse (PSL(2,q)), де q∈{5,7,8,9,11,13}. Основним результатом роботи є доведення того факту, що якщо G є групою, для якої nse (G)=nse(PSL(2,q)), де q∈16,17,19,23, то G≅PSL(2,q)
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