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Abelian Groups With Isomorphic Intersection Graphs  [PDF]
Ergün Yaraneri
Mathematics , 2013,
Abstract: Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct vertices X and Y if and only if X \ Y 6= 1 where 1 denotes the trivial subgroup of G: It was conjectured in [2] that two (non cyclic) finite abelian groups with isomorphic intersection graphs are isomorphic. In this paper we study this conjecture and show that it is almost true. For any inite abelian group D let Dnc be the product of all noncyclic Sylow subgroups of D: Our main result is that: given any two (nontrivial) inite abelian groups A and B; their intersection graphs G(A) and G(B) are isomorphic if and only if the groups Anc and Bnc are isomorphic, and there is a bijection between the sets of (nontrivial) cyclic Sylow subgroups of A and B satisfying a certain condition. So, in particular, two finite abelian groups with isomorphic intersection graphs will be isomorphic provided that one of the groups has no (nontrivial) cyclic Sylow subgroup. Our methods are elementary.
Intersection graph of cyclic subgroups of groups  [PDF]
R. Rajkumar,P. Devi
Mathematics , 2015,
Abstract: Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group $G$, $girth(\mathscr I_c (G)) \in \{3, \infty\}$. Moreover, we classify all finite non-cyclic abelian groups whose intersection graph of cyclic subgroups is planar. Also for any group $G$, we determine the independence number, clique cover number of $\mathscr I_c (G)$ and show that $\mathscr I_c (G)$ is weakly $\alpha$-perfect. Among the other results, we determine the values of $n$ for which $\mathscr I_c (\mathbb{Z}_n)$ is regular and estimate its domination number.
On the automorphisms of a graph product of abelian groups  [PDF]
Mauricio Gutierrez,Adam Piggott,Kim Ruane
Mathematics , 2007,
Abstract: We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature.
Groups in which every non-abelian subgroup is self-centralizing  [PDF]
Costantino Delizia,Heiko Dietrich,Primoz Moravec,Chiara Nicotera
Mathematics , 2015,
Abstract: We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups with the above property.
The subgroup graph of a group
David F. Anderson,Jodi Fasteen,John D. LaGrange
Arabian Journal of Mathematics , 2012, DOI: 10.1007/s40065-012-0018-1
Abstract: Given any subgroup H of a group G, let Γ H (G) be the directed graph with vertex set G such that x is the initial vertex and y is the terminal vertex of an edge if and only if x ≠ y and ${xy\in H}$ . Furthermore, if ${xy\in H}$ and ${yx\in H}$ for some ${x,y\in G}$ with x ≠ y, then x and y will be regarded as being connected by a single undirected edge. In this paper, the structure of the connected components of Γ H (G) is investigated. All possible components are provided in the cases when |H| is either two or three, and the graph Γ H (G) is completely classified in the case when H is a normal subgroup of G and G/H is a finite abelian group.
Rigidity of graph products of abelian groups  [PDF]
Mauricio Gutierrez,Adam Piggott
Mathematics , 2007,
Abstract: We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.
Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup  [PDF]
Qayum Khan
Mathematics , 2006, DOI: 10.1016/j.jpaa.2008.07.005
Abstract: Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell's unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial group. The broader scope is the study of the L-theory of virtually cyclic groups, based on the Farrell--Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2-group and when S is a special 2-group.
Connectivity of Intersection Graphs of Finite Groups  [PDF]
Sel?uk Kayacan
Mathematics , 2015,
Abstract: The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper, we classify finite solvable groups whose intersection graphs are not $2$-connected and finite nilpotent groups whose intersection graphs are not $3$-connected. Our methods are elementary.
A generating set for the automorphism group of a graph product of abelian groups  [PDF]
Luis Corredor,Mauricio Gutierrez
Mathematics , 2009,
Abstract: We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurence's paper [11].
Noether's problem for $p$-groups with an abelian subgroup of index $p$  [PDF]
Ivo M. Michailov
Mathematics , 2012,
Abstract: Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$-automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational over $K$. Let $p$ be an odd prime and let $G$ be a $p$-group of exponent $p^e$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we prove that $K(G)$ is rational over $K$ for the following two types of groups: {\rm (1)} $G$ is a finite $p$-group with an abelian normal subgroup $H$ of index $p$, such that $H$ is a direct product of normal subgroups of $G$ of the type $C_{p^b}\times (C_p)^c$ for some $b,c:1\leq b,0\leq c$; {\rm (2)} $G$ is any group of order $p^5$ from the isoclinic families with numbers $1,2,3,4,8$ and 9.
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