Rocky Mountain Journal of Mathematics

Dominating sets in intersection graphs of finite groups

Selcuk Kayacan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $G$ be a group. The intersection graph $\Gamma (G)$ of $G$ 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 study the dominating sets in intersection graphs of finite groups. We classify abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related to Burnside rings. We introduce the notion of an intersection graph of a $G$-set (somewhat generalizing the ordinary definition of an intersection graph of a group) and establish a general upper bound for the domination number of $\Gamma (G)$ in terms of subgroups satisfying a certain property in the Burnside ring. The intersection graph of $G$ is the $1$-skeleton of the simplicial complex. We name this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that, if the domination number of $\Gamma (G)$ is 1, then the intersection complex of $G$ is contractible.

Article information

Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2311-2335.

First available in Project Euclid: 14 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20D99: None of the above, but in this section
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C69: Dominating sets, independent sets, cliques 20C05: Group rings of finite groups and their modules [See also 16S34] 55U10: Simplicial sets and complexes

Finite groups subgroup intersection graph dominating sets domination number Burnside ring order complex


Kayacan, Selcuk. Dominating sets in intersection graphs of finite groups. Rocky Mountain J. Math. 48 (2018), no. 7, 2311--2335. doi:10.1216/RMJ-2018-48-7-2311.

Export citation


  • H. Ahmadi and B. Taeri, Planarity of the intersection graph of subgroups of a finite group, J. Alg. Appl. 15 (2016), 1650040.
  • R. Baer, The significance of the system of subgroups for the structure of the group, Amer. J. Math. 61 (1939), 1–44.
  • J. Bosák, The graphs of semigroups, in Theory of graphs and its applications, Publ. House Czechoslov. Acad. Sci., Prague, 1964.
  • B. Brešar, P. Dorbec, W. Goddard, B.L. Hartnell, M.A. Henning, S. Klavžar and D.F. Rall, Vizing's conjecture: A survey and recent results, J. Graph Th. 69 (2012), 46–76.
  • B. Brešar, S. Klavžar, and D.F. Rall, Dominating direct products of graphs, Discr. Math. 307 (2007), 1636–1642.
  • K.S. Brown, Euler characteristics of groups: The $p$-fractional part, Invent. Math. 29 (1975), 1–5.
  • I. Chakrabarty, S. Ghosh, T.K. Mukherjee, and M.K. Sen, Intersection graphs of ideals of rings, Discr. Math. 309 (2009), 5381–5392.
  • J.H.E. Cohn, On $n$-sum groups, Math. Scand. 75 (1994), 44–58.
  • B. Csákány and G. Pollák, The graph of subgroups of a finite group, Czechoslov. Math. J. 19 (1969), 241–247.
  • E. Detomi and A. Lucchini, On the structure of primitive $n$-sum groups, Cubo 10 (2008), 195–210.
  • F.V. Fomin, F. Grandoni and D. Kratsch, Measure and conquer: Domination, A case study, Springer, Berlin, 2005.
  • F. Fumagalli, A characterization of solvability for finite groups in terms of their frame, J. Algebra 322 (2009), 1919–1945.
  • M. Garonzi and A. Lucchini, Covers and normal covers of finite groups, J. Algebra 422 (2015), 148–165.
  • D. Gorenstein, Finite groups, Chelsea Publishing Co., New York, 1980.
  • T. Hawkes, I.M. Isaacs and M. Özaydin, On the Möbius function of a finite group, Rocky Mountain J. Math. 19 (1989), 1003–1034.
  • T.W. Haynes, P.J. Slater and S.T. Hedetniemi, Domination in graphs: Advanced topics, Marcel Dekker, New York, 1997.
  • I. Heckenberger, J. Shareshian and V. Welker, On the lattice of subracks of the rack of a finite group, Trans. Amer. Math. Soc., to appear.
  • S.H. Jafari and N. Jafari Rad, Planarity of intersection graphs of ideals of rings, Int. Elect. J. Algebra 8 (2010), 161–166.
  • S.H. Jafari and N. Jafari Rad, Domination in the intersection graphs of rings and modules, Italian J. Pure Appl. Math. (2011), 19–22.
  • S. Kayacan, Connectivity of intersection graphs of finite groups, Comm. Alg. 46 (2018), 1492–1505.
  • S. Kayacan and E. Yaraneri, Abelian groups with isomorphic intersection graphs, Acta Math. Hungar. 146 (2015), 107–127.
  • ––––, Finite groups whose intersection graphs are planar, J. Korean Math. Soc. 52 (2015), 81–96.
  • J.D. Laison and Y. Qing, Subspace intersection graphs, Discr. Math. 310 (2010), 3413–3416.
  • M.S. Lucido, On the poset of non-trivial proper subgroups of a finite group, J. Alg. Appl. 2 (2003), 165–168.
  • D. Quillen, Homotopy properties of the poset of nontrivial $p$-subgroups of a group, Adv. Math. 28 (1978), 101–128.
  • J.J. Rotman, An introduction to the theory of groups, Grad. Texts Math. 148 (1995).
  • R. Schmidt, Subgroup lattices of groups, de Gruyter Expos. Math. 14 (1994).
  • J. Shareshian and R. Woodroofe, A new subgroup lattice characterization of finite solvable groups, J. Algebra 351 (2012), 448–458.
  • ––––, Order complexes of coset posets of finite groups are not contractible, Adv. Math. 291 (2016), 758–773.
  • R. Shen, Intersection graphs of subgroups of finite groups, Czechoslov. Math. J. 60 (2010), 945–950.
  • S.D. Smith, Subgroup complexes, Math. Surv. Mono. 179 (2011).
  • T. tom Dieck, Transformation groups and representation theory, Lect. Notes Math. 766 (1979).
  • V. Welker, personal communication, January 2016.
  • E. Yaraneri, Intersection graph of a module, J. Alg. Appl. 12 (2013), 1250218.
  • ––––, personal communication, June 2014.
  • B. Zelinka, Intersection graphs of finite abelian groups, Czechoslov. Math. J. 25 (1975), 171–174.