گراف هاي هم - ماكسيمال زير گروه هاي گروه ها

چكيده انگليسي :

Co maximal graphs of subgroups of groups NEGAR NOORBAKHSH n noorbakhsh@math iut ac ir January 13 2019 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Bijan Taeri b taeri@cc iut ac ir2000 MSC 05C25 05E12 20D10 20D15Keywords Co maximal graphs of groups Diameter Meta abelian group metacyclic group nilpotent group Abstract This M Sc thesis is based on the following papers Akbari S Miraftab B and Nikandish R Co maximal graphs of subgroups of groups Canad Math Bull 60 1 2017 12 25 Miraftab B and Nikandish R co maximal graphs of two generator groups Journal of Algebra and Its Applications 2019 Let G be a group The co maximal graph of subgroups of G denoted by G is a graph whose vertices are non trivial and proper subgroups of G and two distinct vertices L and K are adjacent in G if and only if G LK In this M Sc thesis we study the diameter clique number and vertex chromatic number of G For instance ifG be free abelian group then diam G 2 We investigate the connectivity of a co maximal graph of a group We prove that if G 1 then G is a connected graph and diam G 3 where and diam are the minmum degree and the diameter of a graph We also show that for a nilpotent group G the graph G is connected if and only if G 1 or G Zp2 where G is the Frattini subgroup of G and Zn is the cyclic group of order n We characterize all finite groups whose co maximal graphs are connected Among other results we show that if G is a finitely generated solvable group and G is connected and more over the degree of a maximal subgroup is finite then G is finite Furthermore we show that the degree of each vertex in theco maximal graph of a general linear group over an algebraically closed field is zero or infinite Also we completelyanswer the following question when is G tree Moreover we characterize all finitely generated abelian groupswhose co maximal graphs are forests We prove also the following results i Let G be a group If G 1 then every vertex of G is adjacent to a maximal subgroup of finite index and moreover G 1 ii Let G be a metabelian group If G 1 and deg M for a maximal subgroup M then G is a finite group iii Let G be a finitely generated nilpotent group Then G G Max G where and are the chromatic and clique number of a graph and Max G is the set of maximal subgroups of a group G We also consider a graph G on a group G The vertices of G are nontrivial elements of G and two distinctvertices x and y are adjacent in G if and only if x y G We classify all groups G such that G iscomplete We study G when G is abelian groups For instance the chromatic number and clique number of G where G is a finite cyclic group or G is abelian 2 generated groups We show that if G be an infinte abeliangroup and G H where G is not null then G H Also we compute some parameters of G in case the group is non abelian We prove that if G is a finite non cyclic group then G q 1 where q is a prime number with q G

نوربخش، نگار

گراف هاي هم - ماكسيمال زير گروه هاي گروه ها

قطر , گراف هم- ماكسيمال گروه ها , گروه پوچ توان , گروه حل پذير , گروه فراآبلي , گروه فرادوري