Hamiltonian cycles in the generating graphs of finite groups

Abstract

Abstract. For a finite group G let Γ(G) denote the graph defined on the nonidentity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph Γ(G) contains a Hamiltonian cycle for many finite groups G.

On the uniform spread of almost simple linear groups

“…Define the element y as in the proof of Propositions 5.8 or 5.9, according to the parity of n. As before, note that some power of y has order r, where r is a primitive prime divisor of q 2 k 1 − 1, so we can use the main theorem of [24] to restrict the subgroups containing gs. Also note that (2 k 1 , r) = (4, 5) if n ≤ 7, otherwise (2 k 1 , r) = (8,17). As usual, let M denote the set of maximal subgroups of G containing gs.…”

“…Similarly, G has spread 2 if and only if the diameter of Γ(G) is at most 2. An even stronger conjecture is proposed in [8]: if |G| ≥ 4 then Γ(G) contains a Hamiltonian cycle (a path that visits each vertex exactly once) if and only if G/N is cyclic for every nontrivial normal subgroup N of G (see [8,Conjecture 1.6]). For example, it is known that all sufficiently large finite simple groups have this remarkable property (see [8]).…”

“…If = p is the defining characteristic then H must appear in [24, Table 6], but there are no relevant cases. Finally, let us assume = p. By inspecting [24,Tables 7,8], and by considering the corresponding Frobenius-Schur indicators (see [15] and [29]) to determine whether or not H ∩ PGL(V ) fixes an appropriate form on V , we reduce to the following cases:…”

Generating sets of finite groups

“…Such groups G have long been studied by means of the generating graph, whose vertices are the elements of G, the edges being the 2-element generating sets. The generating graph was defined by Liebeck and Shalev in [16], and has been further investigated by many authors: see for example [3,5,6,12,18,19,20,23] for some of the range of questions that have been considered. Many deep structural results about finite groups can be expressed in terms of the generating graph.…”

Alternating and Symmetric Groups With Eulerian Generating Graph

“…As an illustration, recall that a graph is Hamiltonian (respectively, Eulerian) if it contains a cycle going through every vertex (respectively, edge) of exactly once. In [5], Breuer et al and the first author have investigated the finite groups G for which (G) is Hamiltonian. For example they showed that every finite simple group of large enough order has a Hamiltonian generating graph and proposed (in correspondence with [4,Conjecture 1.8]) an interesting conjecture characterizing the finite groups having a Hamiltonian generating graph.…”