TY - JOUR
T1 - The intersection graph of a finite simple group has diameter at most 5
AU - Freedman, Saul D.
N1 - The author was supported by a St Leonard’s International Doctoral Fees Scholarship and a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews.
PY - 2021/2/13
Y1 - 2021/2/13
N2 - Let G be a non-abelian finite simple group. In addition, let ΔG be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of ΔG has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.
AB - Let G be a non-abelian finite simple group. In addition, let ΔG be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of ΔG has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.
KW - Intersection graph
KW - Simple group
KW - Subgroups
U2 - 10.1007/s00013-021-01583-3
DO - 10.1007/s00013-021-01583-3
M3 - Article
VL - First Online
JO - Archiv der Mathematik
JF - Archiv der Mathematik
SN - 0003-889X
ER -