[Paper Review] GAP Computations Concerning Hamiltonian Cycles in the Generating Graphs of Finite Groups

Thomas Breuer|arXiv (Cornell University)|Nov 30, 2009

Finite Group Theory Research4 references3 citations

TL;DR

This paper presents computational evidence using GAP that the generating graphs of finite groups—particularly PSL(2,q), nonabelian simple groups up to order 10^7, and certain nonsolvable groups—contain Hamiltonian cycles. By applying character-theoretic lower bounds on vertex degrees and verifying Pósa’s and Chvátal’s criteria, the authors confirm Hamiltonicity for these groups, supporting a broader conjecture on the existence of Hamiltonian cycles in generating graphs of finite noncyclic groups.

ABSTRACT

This is a collection of examples showing how the GAP system can be used to compute information about the generating graphs of finite groups. It includes all examples that were needed for the computational results in the paper "Hamiltonian cycles in the generating graphs of finite groups" by Thomas Breuer, Robert M. Guralnick, Andrea Lucchini, Attila Maróti, and Gábor P. Nagy. The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the GAP code shown for the examples can be used as test input for automatic checking of the data and the functions used.

Motivation & Objective

• To computationally verify the existence of Hamiltonian cycles in the generating graphs of finite noncyclic groups.
• To test the validity of Pósa’s and Chvátal’s criteria for Hamiltonian cycles using vertex degree bounds.
• To support the conjecture that a finite noncyclic group has a Hamiltonian generating graph if and only if every nontrivial normal subgroup has cyclic quotient.
• To provide reproducible GAP code and data for verification of computational results in prior work.
• To extend results on Hamiltonian cycles to sporadic simple groups, their automorphism groups, and PSL(2,q) for larger q.

Proposed method

• Computing vertex degrees in generating graphs using character-theoretic methods based on conjugacy classes and maximal subgroups.
• Applying the closure criterion to reduce the problem to checking Hamiltonicity in the iterated closure of the graph.
• Using Pósa’s and Chvátal’s sufficient conditions: checking that vertex degrees satisfy d_k ≥ k+1 or d_{m−k} ≥ m−k for k < m/2.
• Implementing lower bounds on vertex degrees via permutation characters 1_M^G and multiset of primitive characters Π.
• Employing GAP functions to compute character tables, primitive permutation characters, and conjugacy class structures for groups.
• Validating results through automated testing using GAP code, with assertions disabled for compatibility.

Experimental results

Research questions

• RQ1 Do the generating graphs of all nonabelian simple groups of order up to 10^7 contain a Hamiltonian cycle?
• RQ2 Does the generating graph of PSL(2,q) admit a Hamiltonian cycle for all prime powers q ≥ 13?
• RQ3 Is the conjecture that a finite noncyclic group has a Hamiltonian generating graph iff G/N is cyclic for all nontrivial normal subgroups N, true for nonsolvable groups of order ≤ 10^6?
• RQ4 Can character-theoretic lower bounds on vertex degrees be effectively used to verify Hamiltonicity without full enumeration?
• RQ5 What is the computational behavior of the closure criterion in generating graphs of finite groups?

Key findings

• For all nonabelian simple groups of order up to 10^7, the generating graph contains a Hamiltonian cycle, as confirmed by GAP computations.
• The generating graph of PSL(2,q) satisfies Pósa’s criterion for all prime powers q > 11, with vertex degrees exceeding the number of elements of order 2, 3, 4, or 5.
• For q > 11, each involution in PSL(2,q) has vertex degree greater than n(q,2) = q^2 - 1 (if q even) or ≤ q(q+1)/2 (if q odd), satisfying Pósa’s condition.
• When φ((q+1)/k) ≥ 12, elements of order 3, 4, or 5 in PSL(2,q) have vertex degrees exceeding the total number of elements of order ≤ 5, satisfying Pósa’s criterion.
• The GAP code successfully verifies Hamiltonicity for PSL(2,q) with 2 ≤ q ≤ 11, even though the degree bounds do not hold, via direct computation of the 0th closure.
• The Monster group has exactly one conjugacy class of maximal subgroups isomorphic to PSL(2,41), and no maximal subgroups with socle PSL(2,27), improving prior lower bounds in Section 4.2.