Browse > Article
http://dx.doi.org/10.4134/JKMS.2015.52.1.081

FINITE GROUPS WHOSE INTERSECTION GRAPHS ARE PLANAR  

Kayacan, Selcuk (Department of Mathematics Istanbul Technical University)
Yaraneri, Ergun (Department of Mathematics Istanbul Technical University)
Publication Information
Journal of the Korean Mathematical Society / v.52, no.1, 2015 , pp. 81-96 More about this Journal
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 characterize all finite groups whose intersection graphs are planar. Our methods are elementary. Among the graphs similar to the intersection graphs, we may count the subgroup lattice and the subgroup graph of a group, each of whose planarity was already considered before in [2, 10, 11, 12].
Keywords
finite groups; subgroup; intersection graph; planar;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. Alonso, Groups of order $pq^m$ with elementary abelian Sylow q-subgroups, Proc. Amer. Math. Soc. 65 (1977), no. 1, 16-18.   DOI
2 D. Gorenstein, Finite Groups, 2nd Edition, Chelsea Publishing Co., New York, 1980.
3 J. P. Bohanon and L. Reid, Finite groups with planar subgroup lattices, J. Algebraic Combin. 23 (2006), no. 3, 207-223.   DOI
4 W. Burnside, Theory of Groups of Finite Order, 2nd Edition, Dover Publications Inc., New York, 1955.
5 F. N. Cole and J.W. Glover, On groups whose orders are products of three prime factors, Amer. J. Math. 15 (1893), no. 3, 191-220.   DOI   ScienceOn
6 O. Holder, Die Gruppen der Ordnungen $p^3$, $pq^2$, pqr, $p^4$, Math. Ann. 43 (1893), no. 2-3, 301-412.   DOI
7 R. Le Vavasseur, Les groupes d'ordre $p^2q^2$, p etant un nombre premier plus grand que le nombre premier q, Ann. Sci. Ecole Norm. Sup. (3) 19 (1902), 335-355.
8 G. A. Miller, Groups having a small number of subgroups, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 367-371.   DOI
9 J. J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate Texts in Mathematics, Vol. 148, Springer-Verlag, New York, 1995.
10 R. Schmidt, On the occurrence of the complete graph $K_5$ in the Hasse graph of a finite group, Rend. Sem. Mat. Univ. Padova 115 (2006), 99-124.
11 R. Schmidt, Planar subgroup lattices, Algebra Universalis 55 (2006), no. 1, 3-12.
12 C. L. Starr and G. E. Turner, III, Planar groups, J. Algebraic Combin. 19 (2004), no. 3, 283-295.   DOI
13 E. Yaraneri, Intersection graph of a module, J. Algebra Appl. 12 (2013), no. 5, 1250218, 30 pp.   DOI