GAP (CTblLibXpls) - Chapter 4: GAP Computations Concerning Hamiltonian Cycles in the Generating Graphs of Finite Groups

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4 GAP Computations Concerning Hamiltonian Cycles in the Generating Graphs of Finite Groups
4.1 Overview
4.2 Theoretical Background
4.2-1 Character-Theoretic Lower Bounds for Vertex Degrees
4.2-2 Checking the Criteria
4.3 GAP Functions for the Computations
4.3-1 Computing Vertex Degrees from the Group
4.3-2 Computing Lower Bounds for Vertex Degrees
4.3-3 Evaluating the (Lower Bounds for the) Vertex Degrees
4.4 Character-Theoretic Computations
4.4-1 Sporadic Simple Groups, except the Monster
4.4-2 The Monster
4.4-3 Nonsimple Automorphism Groups of Sporadic Simple Groups
4.4-4 Alternating and Symmetric Groups \(A_n\), \(S_n\), for \(5 \leq n \leq 13\)
4.5 Computations With Groups
4.5-1 Nonabelian Simple Groups of Order up to \(10^7\)
4.5-2 Nonsimple Groups with Nonsolvable Socle of Order at most \(10^6\)
4.6 The Groups \(PSL(2,q)\)

### 4 GAP Computations Concerning Hamiltonian Cycles in the Generating Graphs of Finite Groups

Date: April 24th, 2012

This is a collection of examples showing how the GAP system [GAP21] 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 [BGL+10].

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.

A first version of this document, which was based on GAP 4.4.12, is available in the arXiv at http://arxiv.org/abs/0911.5589v1 since November 2009. The differences between this file and the current document are as follows.

- The format of the GAP output was adjusted to the changed behaviour of GAP 4.5.

- The records returned by `IsomorphismTypeInfoFiniteSimpleGroup` (Reference: IsomorphismTypeInfoFiniteSimpleGroup) contain a component `"shortname"` since GAP 4.11.

- The sporadic simple Monster group has exactly one class of maximal subgroups of the type \(PSL(2, 41)\) (see [NW13]), and has no maximal subgroups which have the socle \(PSL(2, 27)\) (see [Wil10]). As a consequence, the lower bounds computed in Section 4.4-2 have been improved.

- The known information about the primitive permutation characters of the sporadic simple Monster group is available in the data file `data/prim_perm_M.json` of GAP's library of character tables since the release of CTblLib 1.3.3. The data from this file are used in Section 4.4-2 instead of the explicit list that had been defined in the code in earlier versions.

#### 4.1 Overview

The purpose of this note is to document the GAP computations that were carried out in order to obtain the computational results in [BGL+10].

In order to keep this note self-contained, we first describe the theory needed, in Section 4.2. The translation of the relevant formulae into GAP functions can be found in Section 4.3. Then Section 4.4 describes the computations that only require (ordinary) character tables in the GAP Character Table Library [Bre22]. Computations using also the groups are shown in Section 4.5.

The examples use the GAP Character Table Library and the GAP Library of Tables of Marks, so we first load these packages in the required versions.

    gap> if not CompareVersionNumbers( GAPInfo.Version, "4.5" ) then
    >      Error( "need GAP in version at least 4.5" );
    >    fi;
    gap> LoadPackage( "ctbllib", "1.2", false );
    true
    gap> LoadPackage( "tomlib", "1.1.1", false );
    true

#### 4.2 Theoretical Background

Let \(G\) be a finite noncyclic group and denote by \(G^{\times}\) the set of nonidentity elements in \(G\). We define the generating graph \(\Gamma(G)\) as the undirected graph on the vertex set \(G^{\times}\) by joining two elements \(x, y \in G^{\times}\) by an edge if and only if \(\langle x, y \rangle = G\) holds. For \(x \in G^{\times}\), the vertex degree \(d(\Gamma, x)\) is \(|\{ y \in G^{\times}; \langle x, y \rangle = G \}|\). The closure \(cl(\Gamma)\) of the graph \(\Gamma\) with \(m\) vertices is defined as the graph with the same vertex set as \(\Gamma\), where the vertices \(x, y\) are joined by an edge if they are joined by an edge in \(\Gamma\) or if \(d(\Gamma, x) + d(\Gamma, y) \geq m\). We denote iterated closures by \(cl^{(i)}(\Gamma) = cl(cl^{(i-1)}(\Gamma))\), where \(cl^{(0)}(\Gamma) = \Gamma\).

In the following, we will show that the generating graphs of the following groups contain a Hamiltonian cycle:

- Nonabelian simple groups of orders at most \(10^7\),

- groups \(G\) containing a unique minimal normal subgroup \(N\) such that \(N\) has order at most \(10^6\), \(N\) is nonsolvable, and \(G/N\) is cyclic,

- sporadic simple groups and their automorphism groups.

Clearly the condition that \(G/N\) is cyclic for all nontrivial normal subgroups \(N\) of \(G\) is necessary for \(\Gamma(G)\) being connected, and [BGL+10, Conjecture 1.6] states that this condition is also sufficient. By [BGL+10, Proposition 1.1], this conjecture is true for all solvable groups, and the second entry in the above list implies that this conjecture holds for all nonsolvable groups of order up to \(10^6\).

The question whether a graph \(\Gamma\) contains a Hamiltonian cycle (i. e., a closed path in \(\Gamma\) that visits each vertex exactly once) can be answered using the following sufficient criteria (see [BGL+10]). Let \(d_1 \leq d_2 \leq \cdots \leq d_m\) be the vertex degrees in \(\Gamma\).

Pósa's criterion:

  If \(d_k \geq k+1\) holds for \(1 \leq k < m/2\) then \(\Gamma\) contains a Hamiltonian cycle.

Chvátal's criterion:

  If \(d_k \geq k+1\) or \(d_{m-k} \geq m-k\) holds for \(1 \leq k < m/2\) then \(\Gamma\) contains a Hamiltonian cycle.

Closure criterion:

  A graph contains a Hamiltonian cycle if and only if its closure contains a Hamiltonian cycle.

##### 4.2-1 Character-Theoretic Lower Bounds for Vertex Degrees

Using character-theoretic methods similar to those used to obtain the results in [BGK08] (the computations for that paper are shown in [Breb]), we can compute lower bounds for the vertex degrees in generating graphs, as follows.

Let \(R\) be a set of representatives of conjugacy classes of nonidentity elements in \(G\), fix \(s \in G^{\times}\), let \(𝕄(G,s)\) denote the set of those maximal subgroups of \(G\) that contain \(s\), let \(𝕄(G,s)/\sim\) denote a set of representatives in \(𝕄(G,s)\) w. r. t. conjugacy in \(G\). For a subgroup \(M\) of \(G\), the permutation character \(1_M^G\) is defined by

\[ 1_M^G(g):= (|G| \cdot |g^G \cap M|) / (|M| \cdot |g^G|), \]

where \(g^G = \{ g^x; x \in G \}\), with \(g^x = x^{-1} g x\), denotes the conjugacy class of \(g\) in \(G\). So we have \(1_M^G(1) = |G|/|M|\) and thus \(|g^G \cap M| = |g^G| \cdot 1_M^G(g) / 1_M^G(1)\).

Doubly counting the set \(\{ (s^x, M^y); x, y \in G, s^x \in M^y \}\) yields \(|M^G| \cdot |s^G \cap M| = |s^G| \cdot |\{ M^x; x \in G, s \in M^x \}|\) and thus \(|\{ M^x; x \in G, s \in M^x \}| = |M^G| \cdot 1_M^G(s) / 1_M^G(1) \leq 1_M^G(s)\). (If \(M\) is a maximal subgroup of \(G\) then either \(M\) is normal in \(G\) or self-normalizing, and in the latter case the inequality is in fact an equality.)

Let \(\Pi\) denote the multiset of primitive permutation characters of \(G\), i. e., of the permutation characters \(1_M^G\) where \(M\) ranges over representatives of the conjugacy classes of maximal subgroups of \(G\).

Define

\[ \delta(s, g^G):= |g^G| \cdot \max\left\{ 0, 1 - \sum_{{\pi \in \Pi}} \pi(g) \cdot \pi(s) / \pi(1) \right\} \]

and \(d(s, g^G):= |\{ x \in g^G; \langle s, x \rangle = G \}|\), the contribution of the class \(g^G\) to the vertex degree of \(s\). Then we have \(d(\Gamma(G), s) = \sum_{{x \in R}} d(s, x^G)\) and

d(s, g^G) | = | |g^G| - |⋃_{M ∈ M(G,s)} { x ∈ g^G; ⟨ x, s ⟩ ⊆ M }|
|  | ≥ | |g^G| ⋅ max{ 0, 1 - Σ_{M ∈ M(G,s)} 1_M^G(g) / 1_M^G(1) }
| = | |g^G| ⋅ max{ 0, 1 - Σ_{M ∈ M(G,s)} 1_M^G(g) / 1_M^G(1) }
| ≥ | |g^G| ⋅ max{ 0, 1 - Σ_{M ∈ M(G,s)/∼} 1_M^G(g) ⋅ 1_M^G(s) / 1_M^G(1) }
| = | δ(s, g^G)

So \(\delta(s):= \sum_{x \in R} \delta(s, x^G)\) is a lower bound for the vertex degree of \(s\); this bound can be computed if \(\Pi\) is known.

For computing the vertex degrees of the iterated closures of \(\Gamma(G)\), we define \(d^{(0)}(s, g^G):= d(s, g^G)\) and

d^(i+1)(s,g^G):= |g^G| if d^(i)(Г(G),s) + d^(i)(Г(G),g) ≥ |G|-1, and d^(i+1)(s,g^G):= d^(i)(s,g^G) otherwise. Then d(&cl;^{(i)}(\Gamma(G)), s) = \sum_{{x \in R}} d^{(i)}(s, g^G) holds.

Analogously, we set \delta^{(0)}(s, g^G):= \delta(s, g^G),