Divisibility Graph for Symmetric and Alternating Groups

Abstract

Let X be a nonempty set of positive integers and X* = X∖{1}. The divisibility graph D(X) has X* as the vertex set, and there is an edge connecting a and b with a, b ∈ X* whenever a divides b or b divides a. Let X = cs(G) be the set of conjugacy class sizes of a group G. In this case, we denote D(cs(G)) by D(G). In this paper, we will find the number of connected components of D(G) where G is the symmetric group S _n or is the alternating group A _n .

Key Words:

• Alternating group
• Connected component
• Diameter
• Divisibility graph
• Symmetric group

2010 Mathematics Subject Classification:

• 05C25
• 20E45

ACKNOWLEDGMENT

The authors would like to thank the anonymous referee for helpful comments, which improved the quality of this paper.

Notes

Communicated by D. Macpherson.