Finite groups with planar subgroup lattices
Joseph P. Bohanon1
and Les Reid2
1Washington University Department of Mathematics St. Louis Missouri 63130
2Missouri State University Department of Mathematics Springfield Missouri 65897
2Missouri State University Department of Mathematics Springfield Missouri 65897
DOI: 10.1007/s10801-006-7392-8
Abstract
It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three “sporadic” groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.
Pages: 207–223
Keywords: keywords graph; subgroup graph; planar; lattice-planar; nonabelian group
Full Text: PDF
References
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2. J.P. Bohanon, The Planarity of Hasses Lattices of Finite Groups, Master's Thesis, Southwest Missouri State University, 2004.
3. W. Burnside, Theory of Groups of Finite Order, Dover Publications, Cambridge, 1955.
4. D.A. Cox, Primes of the Form x2 + ny2, John Wiley, New York, 1989.
5. D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.
6. D.S. Dummit and R.M. Foote, Abstract Algebra, 3rd ed., John Wiley and Sons, 2004.
7. H.-L. Lin, “On groups of order p2q, p2q2,” Tamkang J. Math. 5 (1974), 167-190.
8. C.R. Platt, “Planar lattices and planar graphs,” J. Combinatorial Theory Ser. B 21 (1976), 30-39.
9. F.S. Roberts, Applied Combinatorics, Prentice-Hall, Englewood Cliffs, NJ, 1984.
10. W.R. Scott, Group Theory, Dover, New York, NY, 1964.
11. C.L. Starr and G. E. Turner III, “Planar groups,” J. Algebraic Combin. 19 (2004), 283-295.
12. J.G. Thompson, “Nonsolvable finite groups All of whose local subgroups are solvable,” Bull. Amer. Math. Soc. 74 (1968), 383-437.
13. A.E. Western, “Groups of order p3q,” Proc. London Math. Soc. 30 (1899), 209-263.
14. J.B. Wilson, Planar Groups,