A characterization of the prime graphs of solvable groups

Abstract

Let π(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G, denoted Γ G , is the graph with vertex set π(G) with edges {p, q} ∈ E(Γ G ) if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.

Two results on character codegrees

“…The codegree prime graph built on cod(G), that we denote by ∆ cod (G), is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cod(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cod(G). The second topic of this paper concerns the relation of solvable groups with the corresponding prime graph of character codegrees and we obtain several results parallel to the results in [5,6].…”

“…The following three results are about the prime graphs (element order version) of solvable groups. Proposition 3.1 (Theorem 2 of [6]). An unlabeled graph F is isomorphic to the prime graph of some solvable group if and only if its complement F is 3-colorable and triangle-free.…”

“…The element order prime graph (Gruenberg-Kegel graph) of a finite group is the (simple undirected) graph whose vertices are the prime numbers dividing the order of the group, with two vertices being linked by an edge if and only if their product divides the order of some element of the group. Gruber et al characterized the element order prime graph of solvable groups only using graph theoretical terms in [6]. Recently Florez et al gave complete characterizations for the element order prime graphs of several classes of groups, such as groups of square-free order, meta-nilpotent groups, groups of cube-free order, and, for any n ∈ N, solvable groups of n th -power-free order in [5].…”

Finite groups with prime graphs of diameter 5

“…The complement of the prime graph of a solvable group does not contain triangles that have been proved by Lucido [3]. In [1] were completely classified of a prime graphs of solvable groups. In particular, the graph Γ is isomorphic to Γ(G), where G is a solvable group if and only if the complement of Γ is without triangles and 3-colorable.…”

Inverse graphs associated with finite groups

“…For an excellent survey on the Hamiltonian cycles in Cayley graphs see [17]. Aside this, several other graphs have also been associated with finite groups such as commuting graphs, intersection graphs, prime graphs, non-commuting graphs, conjugacy class graphs, etc, [5,6,11,12,18]. In order to justify our claim that the inverse graph is new, we as well illustrate by examples how it is different from some known graphs associated with groups.…”