Divisibility Graph for Symmetric and Alternating Groups

Abstract

Abstract. Let X be a non-empty 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

The Divisibility Graph of finite groups of Lie type

“…If |s| is odd, then some power of s b has odd prime order and thus s ∼ s b ∼ x as xs = sx and gcd(|s|, |x|) = 1 by Lemma 4 (1). By Lemma 4(3) s b and thus also s is equivalent to x.…”

“…They found some relationships between the combinatorial properties of D(X), Γ(X) and ∆(X) such as the number of connected components, diameter and girth (see [3,Lemma 1]) and found a relationship between D(X × Y ) and product of D(X) and D(Y ) in [3]. They examined the Divisibility Graph D(G) of a finite group G in [1] and showed that when G is the symmetric or alternating group, then D(G) has at most two or three connected components, respectively. In both cases, at most one connected component is not a single vertex.…”

On divisibility graph for simple Zassenhaus groups

“…The divisibility graph D(G), where G is a symmetric group or an alternating group studied in [1]. It is shown there that D(S n ) has at most two connected components and if it is not connected then one of its connected components is K 1 .…”