A characterization of the prime graphs of solvable groups (original) (raw)

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. 1 Introduction. Prime graphs originated in the 1970s as a by-product of certain cohomological questions posed by K.W. Gruenberg. Shortly after their introduction, prime graphs became objects of interest in their own right, and since then numerous contributions have been made to the topic. The prime graphs of finite simple groups are well understood (see [14], [8], and [15]); as is the structure of groups with acyclic prime graphs (see [10]). Graph invariants such as diameter [11] and degree sequence [12] have also been extensively documented. The question of how graph theoretic properties influence group structure remains, for the most part, open, and it is from this angle that our investigation proceeds. Solvable groups possess several properties that motivate an extended discussion of their prime graphs. Philip Hall established in [2] that G is solvable if and only if G contains a Hall π-subgroup for every subset π ⊂ π(G). Graph theoretically, we can interpret this as the statement that, whenever G is solvable, every induced subgraph Γ G [π] is the prime graph of a Hall π-subgroup of G. Of further use is the following proposition, which we refer to as Lucido's Three Primes Lemma. Lemma 1 (Lucido's Three Primes Lemma, [11]). Let G be a finite solvable group. If p, q, r are distinct primes dividing |G|, then G contains an element of order the product of two of these three primes. Equivalently, if G is solvable, then Γ G cannot contain an independent set of size 3. In other words, Γ G must be triangle free. Williams observed in [15] that every solvable group with a disconnected prime graph must be either a Frobenius or 2-Frobenius group. It follows that whenever an edge pq is missing from the prime graph of a solvable group, the corresponding Hall {p, q}-subgroup H pq admits a fixed point free action between either the Sylow subgroups of H pq or their image in its Fitting quotient. Our characterization begins by defining an acyclic orientation of Γ G indicating the direction of this action for every edge pq ∈ Γ G. We refer to this as the Frobenius digraph of G. The Frobenius digraph affords us a powerful tool for understanding the prime graphs of solvable groups. The primary result of this paper can be summarized by the following theorem, though in fact both implications are derived from stronger results. Theorem 2. A graph F is the prime graph of some solvable group if and only if its complement F is 3-colorable and triangle free. Our characterization has some interesting applications. For example, we find that the girth of the prime graph of a solvable group is exactly equal to three with only a few exceptions, which we then classify.