††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Impact of Monoatomic Vacancies in 2D Materials on the Performance of
Magnetic Tunnel Junction Devices: Insights from Configurations and Interface
Interactions

Halimah Harfah  Corresponding author<EMAIL_ADDRESS>Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-
Cho, Toyonaka, Osaka 560-0043, Japan  Yusuf Wicaksono  RIKEN Cluster for
Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  Gagus
Ketut Sunnardianto  Research Center for Quantum Physics, National Research and
Innovation Agency of Indonesia (BRIN), Tangerang Selatan, Banten, 15314,
Indonesia  Research Collaboration Center for Quantum Technology 2.0, National
Research and Innovation Agency of Indonesia (BRIN), Bandung 40132, Indonesia
School of Materials Science and Engineering, Nanyang Technological University,
50 Nanyang Avenue, Singapore 639798, Singapore  Muhammad Aziz Majidi
Department of Physics, Faculty of Mathematics and Natural Sciences,
Universitas Indonesia, Kampus UI Depok, Depok 16424, Indonesia.  Koichi
Kusakabe  School of Science, Graduate School of Science, University of Hyogo
3-2-1 Kouto, Kamigori-cho, Ako-gun, 678-1297, Hyogo, Japan.

###### Abstract

We investigate the impact of monoatomic vacancies in 2D materials on the
performance of magnetic tunnel junction (MTJ) devices using first-principles
calculations within Density Functional Theory (DFT). Specifically, we analyze
the influence on hexagonal boron nitride (hBN) with various layer
configurations, uncovering distinct transmission probability patterns.
Transmission calculations were conducted using the Landauer-Büttiker formula
employing the Non-Equilibrium Green’s Function (NEGF) method. In the
Ni/hBN(VB)-hBN/Ni system, a significant reduction in transmission probability
was observed compared to non-vacancy configurations. However, when two hBN
vacancies were considered, creating the Ni/hBN(VB)-hBN(VB)/Ni MTJ system, a
new transmission channel mediated by vacancy localized states emerged. The
introduction of a monoatomic boron vacancy in the middle hBN layer of the
Ni/3hBN/Ni system revealed nuanced effects on the transmission probability,
highlighting alterations in the spin minority and majority channels.
Additionally, we explore the monoatomic vacancy in the graphene layer in the
Ni/hBN-Gr-hBN/Ni MTJ, uncovering a unique transmission channel influenced by
the proximity effect. Our findings suggest that the creation of monoatomic
vacancies on the insulator barrier of 2D materials induces distinctive
characteristics shaped by the interaction between the surface state of the
electrode and the localized state of the monoatomic vacancy layer in the MTJ
system.

††preprint: APS/123-QED

##  I Introduction

Magnetic tunnel junctions (MTJs) have attracted significant attention in the
realm of spintronics, owing to their diverse array of potential applications
ranging from logic devices to magnetic sensors [1, 2, 3, 4, 5, 6]. A critical
factor in optimizing MTJ performance is the tunneling magnetoresistance (TMR)
ratio, wherein MgO has emerged as the predominant choice for the tunnel
barrier material. For instance, the CoFeB/MgO/CoFeB MTJ configuration has
showcased remarkable TMR values, reaching as high as 1100% at 4.2 K [7].
However, the pursuit of downsizing these devices by reducing the barrier
thickness has often been impeded by a concomitant decrease in TMR, sometimes
plummeting to 55%, primarily attributed to the presence of uncontrollable
defects within the MgO tunnel barrier [8, 9].

Conversely, researchers have been actively exploring alternatives to MgO,
seeking to replace it with 2D materials as a means of device miniaturization.
The potential of 2D materials such as graphene (Gr) or hexagonal boron nitride
(hBN) as MTJ spin valves has been thoroughly investigated [10, 11, 12, 13, 14,
15, 16, 17]. One such approach involves employing a monolayer or a few layers
of hBN as the tunnel barrier, sandwiched between ferromagnetic electrodes to
form MTJ structures [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In the
Ni/monolayer hBN/Ni MTJ system, the interaction between Ni and N atoms at the
interface can lead to $pd$-hybridization, allowing electrons at the Fermi
energy to easily pass through the monolayer hBN as a tunnel barrier [13]. This
electron transmission is primarily due to propagating-wave electrons. In
contrast, when multilayer hBN serves as the tunnel barrier, the transmission
of electrons through the hBN insulator barrier is facilitated by the Ni
d${}_{z^{2}}$-orbital surface state and the proximity effect induced on the
hBN layer [14]. This finding emphasizes the pivotal role played by the
proximity effect within 2D materials in modulating electron tunneling
capacity. Interestingly, unlike conventional thin-film materials such as MgO,
modified 2D materials, engineered through the introduction of monoatomic
vacancies, exhibit a spectrum of unique physical and chemical properties [29,
30, 31, 32]. These distinct characteristics may mediate novel transmission
phenomena not observed in MgO-based MTJs.

In the field of spintronics, investigating the impact of monoatomic vacancies
in two-dimensional (2D) materials on the performance of MTJ devices is of
critical importance [33, 34, 35, 36]. To understand how single-atom defects
(monoatomic vacancies) in two-dimensional (2D) materials affect the
transmission properties of MTJs, we use a theoretical framework based on
Density Functional Theory (DFT). This allows us to explore the complex
relationship between the electronic properties and device performance. We
examine the transmission properties using the Landauer-Büttiker formula. The
Non-Equilibrium Green’s Function (NEGF) method is then employed to gain
insights into the transport properties of the systems. We focused on hexagonal
boron nitride (hBN), a well-known insulator, and explored how different layer
arrangements within the MTJ device affect its performance. By controlling
monoatomic vacancies and altering the layer structures, we observed
significant changes in the transmission probability, especially in the spin-
dependent channels. Furthermore, we incorporated graphene layers within the
MTJ design and investigated how monoatomic vacancies impact the transmission
properties in these hybrid systems. This study deepens our understanding of
the fundamental mechanisms governing electron transport in MTJ devices and
provides practical guidance for designing and optimizing future spintronic
devices. By investigating the impact of monoatomic vacancies in 2D materials
on device performance, we aim to advance spintronics towards devices with
enhanced functionality and efficiency.

##  II Computational Method

Figure 1:  A $3\times 3$ unit cell of (a) hBN with a monoatomic boron vacancy
(VB), (b) graphene (Gr) with a vacancy, (c) hBN(VB)/hBN, (d) hBN(VB)/hBN(VB),
(e) hBN/hBN(VB)/hBN, and (f) hBN/Gr(V)/hBN used in the calculations. Blue,
black, and pink color balls represent N, C, B atoms, respectively. (g) The
setup to calculate the tunneling transmission probability of the
aforementioned tunnel barrier where Ni is used as electrodes. The scattering
region comprises a three-layer Ni/tunnel barrier/three-layer Ni, and the left
and right electrodes comprise six Ni layers. Visualization was performed using
XCrySDen [37]

In this work, we used a $3\times 3$ supercell to model the MTJ systems. To
introduce defects, a monoatomic boron vacancy (VB) was deliberately created
within the hBN layer, as depicted in Figure 1(a). Additionally, a monoatomic
vacancy was introduced into the graphene layer, as illustrated in Figure 1(b).
We explored various stacking configurations of hBN(VB) layers, including
hBN(VB)/hBN, hBN(VB)/hBN(VB), and hBN/hBN(VB)/hBN, as shown in Figures 1(c),
(d), and (e), respectively. Furthermore, we investigated the hBN/Gr/hBN
stacking with different vacancy configurations, where the vacancy was
positioned on the graphene layer, as depicted in Figure 1(f). The study
primarily focused on evaluating the significance of localized states in spin-
dependent electron tunneling between two ferromagnetic electrodes, with nickel
chosen as the electrode material. Given that the anti-parallel configuration
(APC) is known to exhibit low transmission probability due to the spin-
blocking effect, our analysis primarily focused on the parallel configuration
(PC) state of the MTJ, where the magnetic moments of the upper and lower Ni
electrodes were oriented parallelly upward.

The SIESTA package [38, 39] was utilized for various calculations,
encompassing structural equilibrium determination, assessment of magnetic
properties, mapping of spin-charge density, and determination of the local
density of states (LDOS) utilizing spin-polarized DFT. Electron-ion
interaction within the generalized gradient approximation (GGA) was modeled
using the Troullier–Martins [40] pseudopotential and the
Perdew–Burke–Ernzerhof functional, specifically the PBESol functional [41]. A
basis set incorporating double-zeta and polarization was employed [42, 43,
44]. The atomic positions were relaxed with a force tolerance of 0.01 eV/Å,
and a Monkhorst–Pack k-mesh of $36\times 36\times 1$ was utilized for
calculations, with a mesh cutoff of 500 Ry. Additionally, van der Waals
interactions between hBN and graphene were accounted for by integrating a
Grimme-type dispersion potential [45].

The tunneling transmission probability was computed using the Landauer-
Büttiker formalism within the NEGF method. The setup for calculating the
transmission probability is depicted in Figure 1(g). The spin-dependent
current was determined utilizing the Landauer-Büttiker equation:

$I^{\uparrow(\downarrow)}=\frac{e}{h}\int_{-\infty}^{\infty}T^{\uparrow(\downarrow)}(E)\Big{[}f_{L}(E,\mu)-f_{R}(E,\mu)\Big{]}dE,$
(1)

where $f_{L}(E,\mu)\big{(}f_{R}(E,\mu)\big{)}$ represents the right (left)
moving electrons injected from the left (right) electrode, and
$\mu_{L}\big{(}\mu_{R}\big{)}$ denotes the chemical potentials of the left
(right) electrodes, both assumed to be at the Fermi level ($E_{F}$) due to the
zero bias voltage. The transmission probability, $T$, as a function of energy,
$E$, is described by the Green’s function:

$T^{\uparrow(\downarrow)}(E)=\Tr{\Big{[}\Gamma_{L}G^{R}\Gamma_{R}G^{A}\Big{]}},$
(2)

where $\Gamma_{L}(\Gamma_{R})$ represents the coupling matrix of the left
(right) electrode, and $G^{R}(G^{A})$ denotes the retarded (advanced) Green’s
function of the central region.

##  III Results and Discussion

###  III.1 Influence of Monoatomic Boron Vacancy in One or Two hBN in
Ni/2hBN/Ni MTJ

Figure 2:  (a) The $3\times 3$ Ni/hBN/hBN(VB)/Ni SCDM (red color represents
spin-up electron density). Visualization was performed using XCrySDen [37].
(b) LDOS for Ni-upper, hBN-upper, hBN-lower (hBN with VB), and Ni-lower layers
in PC configuration. The positive (negative) value of DOS represents the spin
majority (minority) channel. (c) A zoom on the lower energy of LDOS in (b).
Figure 3:  (a) The $3\times 3$ Ni/hBN(VB)/hBN(VB)/Ni SCDM (red color
represents spin-up electron density). Visualization was performed using
XCrySDen [37]. (b) LDOS for Ni-upper/lower and hBN-upper/lower (which both
have VB) in the PC configuration. The positive (negative) value of DOS
represents the spin majority (minority) channel. (c) A zoom on the lower
energy of LDOS in (b).

To examine the impact of a monoatomic boron vacancy in hBN on the
functionality of an MTJ device, we investigated two potential configurations:
Ni/hBN(VB)-hBN/Ni and Ni/hBN(VB)-hBN(VB)/Ni MTJ systems. The presence of a
monoatomic boron vacancy induces magnetization in the hBN layer. This
phenomenon is attributed to the unpaired electron of the nitrogen atoms at the
vacancy site, which fails to form a $\sigma$-bond with the boron atom. We
observe vacancy-induced magnetization in both Ni/hBN(VB)-hBN/Ni and
Ni/hBN(VB)-hBN(VB)/Ni MTJs, as depicted in Figures 2(a) and 3(a),
respectively, through spin charge density mapping (SCDM). Notably, the SCDM
reveals dominant spin-up electron density in the localized state near VB,
aligned parallel to the spin-up magnetization direction of the Ni slabs. This
alignment signifies a strong ferromagnetic coupling between the magnetic
moment at the vacancy and the Ni slabs. Furthermore, a short-range magnetic
interaction between the localized state and neighboring nitrogen atoms is
evident from the damping of spin-up electron density amplitude. This damping
occurs in regions where spin-up electron density concentrates near the vacancy
but decreases for nitrogen atoms farther away. The emergence of vacancy-
induced magnetization in hBN(V$B$) is expected to alter its electronic
structure and the propagation of Ni d${z^{2}}$-orbital surface state through
the insulator barrier of two hBN layers.

Initially, we examined the influence of VB on the electronic structure of the
MTJ system using Ni/hBN(VB)-hBN/Ni. This configuration allowed for a direct
comparison between pristine hBN, hybridized with the upper Ni slab, and
hBN(VB), hybridized with the lower Ni slab. Figure 2(b) displays the local
density of states (LDOS) of the upper and lower Ni slabs, pristine upper hBN,
and lower hBN with VB in the PC configuration. A notable difference in LDOS
between the upper and lower Ni slabs is observed, stemming from the
modification of $pd$-hybridization at the interface of hBN(VB)/Ni. This
modification is evidenced by the DOS peak at $E-E_{F}=0.4$ eV in the spin
minority channel, corresponding to hybridization at the Ni/hBN interface
between the Ni d${z^{2}}$ orbital and the nitrogen pz orbital. This peak also
appears in the spin majority channel at approximately $E-E_{F}=-1.0$ eV,
consistent with previous findings[14]. However, in the lower Ni slab,
hybridized with hBN(VB), the DOS peak in the spin minority channel at
$E-E_{F}=0.4$ eV is reduced due to the electron localization at the vacancy
site. A similar reduction is observed in the spin majority channel around
$E-E_{F}=-1.0$ eV. The disparity is more pronounced between the upper and
lower hBN layers, as shown in Figure 2(c). In the upper hBN, a nearly
insulating gap dominates the DOS, with a peak only appearing at $E-E_{F}=0.4$
eV for the spin minority channel and around $E-E_{F}=-1.0$ eV for the spin
majority channel, arising from $pd$-hybridization. Conversely, the lower hBN
containing VB exhibits a DOS peak in the same energy range as the upper hBN,
along with additional states. In the spin majority channel, a higher DOS peak
is observed from $E-E_{F}=-0.9$ to $-0.2$ eV, with the highest and second-
highest DOS at $E-E_{F}=-0.38$ and $-0.65$ eV, respectively. Meanwhile, in the
spin minority channel, a broad range of new states emerges from $E-E_{F}=-1.0$
to $1.0$ eV. The higher occupancy of states in the spin majority channel
compared to the minority channel corresponds to the spin-up magnetization of
hBN(VB) due to the vacancy.

The modification of the LDOS in the lower Ni and hBN layers directly affects
the probability of electron transmission. As previously mentioned, in a two-
layer hBN MTJ system, electron transmission primarily occurs through the Ni
surface state. Figure 4 illustrates a prominent transmission peak at
$E-E_{F}=0.38$ eV ($-0.98$ eV) for spin-down (spin-up) electrons, consistent
with previous findings. This high transmission is attributed to electron
transmission through the Ni d${}_{z^{2}}$ orbital surface state. However, in
the Ni/hBN/hBN(V$B$)/Ni MTJ structure, the decreased Ni d${z^{2}}$ orbital DOS
leads to a significant reduction in electron transmission for both spin-up and
spin-down electrons. Conversely, in the Ni/hBN(VB)-hBN(VB)/Ni MTJ, identical
LDOS in both the upper and lower Ni and hBN layers due to the creation of
vacancies on both hBN layers, as shown in Figure 3(b) and (c), similarly
reduces electron transmission probability for spin-down (spin-up) electrons at
$E-E_{F}=0.38$ eV ($-0.98$ eV) while creating a new transmission channel for
spin-down (spin-up) electrons from $E-E_{F}=0.5$ eV ($-0.85$ eV) to $0.8$ eV
($-0.4$ eV). This suggests that the newly created states in hBN(V$B$) mediate
electron transmission not only for the Ni d${z^{2}}$ orbital surface state but
also for another orbital surface state.

We note that the Ni/hBN(VB)-hBN(VB)/Ni MTJ introduces a new transmission
channel when the hBN(VB) layers are in close proximity. However, as the
separation between hBN(VB) layers increases, the intensity of the transmission
channel decreases and eventually disappears, resembling the transmission
observed in Ni/hBN(VB)-hBN/Ni.

Figure 4:  Comparison of transmission probability for Ni/2hBN/Ni,
Ni/hBN(VB)/hBN/Ni, and Ni/hBN(VB)/hBN(VB)/Ni MTJs.

###  III.2 Interaction Between Ni’s Surface State at Interface with Localized
State of 2D Materials Vacancy

In the preceding section, we examined the influence of VB on the electronic
structure of both the hBN and Ni layers at the Ni/hBN interface. Here, we
delve into the interaction between the surface state of Ni at the interface,
which remains unaltered, and the isolated localized state of the middle hBN
layer in the Ni/hBN/hBN(VB)/hBN/Ni MTJ.

Figure 5(a) illustrates the spin charge density mapping (SCDM) of the
Ni/hBN/hBN(VB)/hBN/Ni MTJ, indicating that the hBN(VB) in the middle possesses
an induced magnetic moment. A strong magnetic moment is observed at the
vacancy site, aligned parallel to the Ni slab, suggesting magnetic interaction
between the hBN(VB) and Ni slab despite lacking a direct interface. The local
density of states (LDOS) of the Ni slab, hBN at the interface, and hBN(VB) is
depicted in Figures 5(b) and (c). The LDOS analysis reveals that the hBN(VB)
in the middle does not alter the DOS of the Ni layer at the interface or the
hBN at the interface. However, the DOS of the hBN(V$B$) exhibits spin
polarization, evident from the Stoner gap. The newly created state from the
vacancy shows some overlap with the surface state of the Ni d${z^{2}}$ orbital
at $E-E_{F}=0.42$ eV ($-0.9$ eV) for the spin minority (majority) channel.

Figure 5:  (a) The $3\times 3$ Ni/hBN/hBN(VB)/hBN/Ni SCDM (red color
represents spin-up electron density). Visualization was performed using
XCrySDen [37]. (b) Local density of states (LDOS) for Ni-upper/lower and hBN-
upper/lower (hBN without VB), and hBN-middle (hBN with VB) in PC
configuration. The positive (negative) value of DOS represents spin majority
(minority) channel. (c) The zoom on lower energy of LDOS in (b). Figure 6:
(a) The $3\times 3$ Ni/hBN/Gr(V)/hBN/Ni SCDM (red color represents spin-up
electron density). Visualization was performed using XCrySDen [37]. (b) LDOS
for Ni-upper/lower, hBN-upper/lower, and Gr(V) in PC configuration. The
positive (negative) value of DOS represents spin majority (minority) channel.
(c) The zoom on lower energy of LDOS in (b).

The transmission probability of the Ni/hBN/hBN(V$B$)/hBN/Ni MTJ is displayed
in Figure 7(a). No new transmission channel is observed. Instead, an increment
(reduction) in the transmission probability is noted for the spin majority
(minority) channel at $E-E_{F}=-0.9$ eV ($0.42$ eV) compared to the Ni/3hBN/Ni
MTJ. This indicates that the Ni d${z^{2}}$ orbital solely affects the occupied
state of the localized state. The reduction in transmission probability in the
spin minority channel, due to the Ni d${z^{2}}$ orbital surface state, does
not interact with the proximity effect, which exhibits spin-up electron
localization. Conversely, the spin majority channel of the Ni d${z^{2}}$
orbital surface state propagates through the localized state, resulting in an
increment in the transmission probability of spin-up electrons.

In a complementary investigation, we introduced a monoatomic vacancy in the
graphene layer to create the Ni/hBN/Gr(V)/hBN/Ni MTJ. Similar to the hBN(VB)
in Ni/hBN/hBN(VB)/hBN/Ni MTJ, the graphene with a vacancy exhibits an induced
magnetic moment and ferromagnetic interaction with the Ni slab, as shown in
Figure 6(a). Furthermore, the vacancy in graphene does not modify the DOS of
the Ni layer at the interface or the hBN layer at the interface, as depicted
in Figures 6(b) and (d). Interestingly, a decrease in spin-down electron
transmission at higher energy, akin to the Ni/hBN/hBN(VB)/hBN/Ni MTJ system,
is observed. In the Ni/hBN/Gr(V)/hBN/Ni MTJ, the vacancy in graphene creates a
new transmission channel for both spin-up and spin-down electrons within the
same energy range, around $E-E_{F}=-0.6$ eV, as shown in Figure 7(b). This
unique transmission channel originates from the proximity effect, which
interacts with the localized state as well as the $\pi$-orbital of the
graphene layer. However, this interaction occurs at the occupied state,
implying that the surface state not only propagates through Gr(V) mediated by
the localized state but can also be redistributed within the graphene layer
through graphene’s $\pi$-orbital. We anticipate the tunneling
magnetoresistance (TMR) ratio at $E-E_{F}=-0.6$ eV to exceed 1200%, as found
in Ni/hBN/Gr/hBN/Ni.

Figure 7:  Transmission probability comparison between (a) Ni/3hBN/Ni and
Ni/hBN/hBN(VB)/hBN/Ni MTJs, and between (b) Ni/hBN/Gr/hBN/Ni and
Ni/hBN/Gr(V)/hBN/Ni MTJs.

##  IV Conclusion

In this study, we investigated the impact of a monoatomic vacancy in 2D
materials on the performance of an MTJ device. Initially, we explored bilayer
hexagonal boron nitride (hBN) configurations, considering Ni/hBN(VB)-hBN/Ni
and Ni/hBN(VB)-hBN(VB)/Ni MTJ systems. The transmission probability was
notably reduced in the Ni/hBN(VB)-hBN/Ni configuration compared to the
pristine bilayer hBN system. Conversely, in the Ni/hBN(VB)-hBN(VB)/Ni
configuration, an additional peak in the transmission probability emerged,
indicating a new transmission channel facilitated by the localized state of
hBN(VB)-hBN(VB).

Subsequently, we introduced a monoatomic boron vacancy in the middle hBN layer
of the Ni/3hBN/Ni system, creating a Ni/hBN-hBN(VB)-hBN/Ni MTJ. The presence
of the localized state in the middle hBN layer led to a reduction
(enhancement) in the transmission probability peak for the spin minority
(majority) channel at higher (lower) energy, attributed to the stoner gap
created in the middle hBN layer. Consequently, the transmission probability
for the spin-majority channel was augmented.

Finally, we investigated the monoatomic vacancy in the graphene layer of the
Ni/hBN-Gr-hBN/Ni MTJ. The presence of the graphene vacancy induced a new
transmission channel for both the spin-majority and minority channels within
the same energy range. This unique transmission channel originated from the
proximity effect interacting with the localized state of the graphene layer.

The introduction of a monoatomic vacancy on the insulator barrier of 2D
materials in the MTJ system results in unique characteristics determined by
the interaction between the surface state of the electrode and the localized
state of the monoatomic vacancy layer. This interaction offers promising
avenues for tailoring the properties and performance of spintronic devices
based on MTJs.

## Conflicts of interest

“There are no conflicts to declare”.

## Acknowledgements

Calculations were performed at the Kyushu University computer center. H.H.
gratefully acknowledges the fellowship support from the JSPS. This study was
partly supported by the Japan Society for the Promotion of Science (JSPS)
KAKENHI (Grant Nos. JP19H00862 and JP16H00914 in the Science of Atomic Layers,
21J22520 in the Grant-in-Aid for Young Scientists, and JP18K03456).

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# The directed Cayley diameter and the Davenport constant

Réka András  Eötvös Loránd University, Faculty of Informatics, H-1117
Budapest, Pázmány P. sny 1/C., Hungary<EMAIL_ADDRESS>,  Kálmán
Cziszter  Budapest Business University, Markó u. 29-31, 1055 Budapest,
Hungary, ORCID iD: https://orcid.org/0000-0002-8034-836X
<EMAIL_ADDRESS>,  Mátyás Domokos  HUN-REN Alfréd Rényi Institute of
Mathematics, Reáltanoda utca 13-15, 1053 Budapest, Hungary, ORCID iD:
https://orcid.org/0000-0002-0189-8831<EMAIL_ADDRESS>and  István
Szöllősi  Babeş-Bolyai University, Faculty of Mathematics and Computer
Science, str. M. Kogălniceanu, nr. 1, 400084, Cluj-Napoca, Romania, ORCID iD:
https://orcid.org/0000-0001-8632-9760<EMAIL_ADDRESS>
###### Abstract.

The directed Cayley diameter of a finite group is investigated using the
monoid of product-one sequences over the group. This relation is used to
develop an algorithm for computing bounds for the directed Cayley diameter of
a non-abelian finite group, based on a method to compute the so-called
geodesic atoms in the monoid of product-one sequences. The algorithm has been
implemented in GAP, and was applied to determine the directed Cayley diameter
for several non-abelian groups of order at most $39$. Furthermore, the small
and large Davenport constants of all non-abelian groups of order at most $39$
are computed, extending thereby the formerly obtained results on the groups of
order less than $32$. Along the way the directed Cayley diameter of a finite
abelian group is expressed in terms of its invariants.

###### Key words and phrases:

Cayley graph, directed diameter of a group, monoid of product one sequences,
Davenport comstant

###### 2010 Mathematics Subject Classification:

Primary 20D60; Secondary 20M14, 20K01

Partially supported by the Hungarian National Research, Development and
Innovation Office, NKFIH K 138828, K 132002.

##  1\. Introduction

It was proved recently in [10] that if the monoid of product-one sequences
over the finite group $G_{1}$ is isomorphic to the monoid of product-one
sequences over the finite group $G_{2}$, then the group $G_{1}$ is isomorphic
to the group $G_{2}$. It is therefore not surprising that much information on
a finite group $G$ can be read off from the set of atoms in the monoid of
product-one sequences over $G$. The present paper gives further instances of
this principle. We shall focus on the problem of computing the directed Cayley
diameter of a finite group. Note that the directed Cayley diameter is an
obvious upper bound for the undirected Cayley diameter (referred to simply as
the “Cayley diameter”). Conversely, a general explicit bound for the directed
Cayley diameter in terms of the Cayley diameter is proved in [1]. The Cayley
diameter of groups has received much attention in the literature. In
particular, several prominent papers are motivated by the conjecture from [2]
stating that the Cayley diameter of a finite non-abelian simple group $G$ is
bounded by $\log^{c}|G|$ with some absolute constant $c$, see for example
[17], [4], [20], [16], [7]. Other areas where Cayley graphs have relevance are
surveyed in [18].

Recall that the directed Cayley diameter of a finite group $G$ is the maximal
diameter of its Cayley graphs associated with the various irredundant
generating systems of $G$. Therefore, if we want to compute the directed
Cayley diameter of $G$ following naively its definition, then we would need to
go through all the irredundant generating systems of $G$. It is not clear how
one should organize this process. However, our approach avoids this by using
product-one sequences over the group.

The starting point of this paper is the observation [5, Proposition 5.6],
asserting that the large Davenport constant $\mathsf{D}(G)$ of a finite group
$G$ is strictly greater than the diameter of any of the Cayley digraphs of
$G$. In fact this statement can be strengthened by saying that $\mathsf{D}(G)$
is strictly greater than the directed Cayley diameter of any subgroup of $G$
(see Corollary 4.3). By our Lemma 5.5, the maximum of the directed Cayley
diameters of the subgroups of $G$ is one less than $\mathsf{GD}(G)$, the
_geodesic Davenport constant of_ $G$ introduced in Definition 5.4 as the
maximal length of a so-called _geodesic atom_ (see Definition 5.1) in the
monoid $\mathcal{B}(G)$ of product one sequences over $G$. A characterisation
of the geodesic atoms, given in Section 5, together with Lemma 6.2, offer an
algorithm to build up the set of geodesic atoms, using a bottom-up approach
(see Section 7). This algorithm is related to the algorithm given in [5,
Section 6], computing $\mathsf{D}(G)$ and all atoms in $\mathcal{B}(G)$.
Finally, once a list of geodesic atoms is produced, one can read off bounds
for the directed Cayley diameter of $G$ (see Lemma 5.5).

After recalling concepts related to Cayley graphs and the monoid of product-
one sequences in Section 2, we turn to finite abelian groups and determine
their directed Cayley diameter in Theorem 3.2. The material on the relation
between the directed Cayley diameter and the atoms in the monoid of product-
one sequences as well as the algorithms are contained in Sections 4, 5, 6, and
7. The results of our computer calculations are presented in Section 8. The
small and large Davenport constants of the non-abelian groups of order less
than $32$ were computed in [5]. Here we give the Davenport constants of the
remaining non-abelian groups of order at most $39$ as well, and for all non-
abelian groups $G$ with $|G|\leq 39$ we compute also the geodesic Davenport
constant of $G$ (which in several cases yields the directed Cayley diameter of
$G$).

Finally, in Section 9 we review some recent results on the Davenport constants
of finite non-abelian groups.

We note that in a follow-up paper we shall adapt the results of this paper to
the problem of computing the (undirected) Cayley diameter of a finite group.

##  2\. Preliminaries

###  2.1. Directed Cayley graphs

Let $G$ be a finite group (written multiplicatively unless explicitly stated
otherwise) and let $B\subseteq G$ be a generating system of $G$. The _directed
Cayley graph_ $\mathrm{Cay}(G,B)$ of $G$ with respect to the generating system
$B$ is the digraph whose vertices are labeled by the elements of $G$, and the
pair $(g,h)$ is a (directed) edge in $\mathrm{Cay}(G,B)$ if there exists an
element $b\in B$ with $h=gb$. We say that a directed path
$g_{0},g_{1},\cdots,g_{d}$ in $\mathrm{Cay}(G,B)$ is _geodesic_ if its length
$d$ is minimal among the lengths of the directed paths in $\mathrm{Cay}(G,B)$
from $g_{0}$ to $g_{d}$. The _diameter_ $\mathrm{diam}(\mathrm{Cay}(G,B))$ of
$\mathrm{Cay}(G,B)$ is the maximal possible length of a geodesic path in
$\mathrm{Cay}(G,B)$. As the action of $G$ on itself via left multiplication
embeds $G$ into the automorphism group of the digraph $\mathrm{Cay}(G,B)$,
$\mathrm{diam}(\mathrm{Cay}(G,B))$ is the same as the maximal length of a
geodesic path in $\mathrm{Cay}(G,B)$ starting at $1_{G}$. The _directed Cayley
diameter_ of $G$ is defined as

$\mathsf{dcd}(G):=\max\\{\mathrm{diam}(\mathrm{Cay}(G,B))\mid B\subseteq
G\mbox{ generates }G\\}.$

The number $\mathsf{dcd}(G)$ can be introduced also without a reference to
Cayley graphs. Namely, for a subset $B\subseteq G$ and an element $g\in\langle
B\rangle$ (the subgroup of $G$ generated by $B$) we set

$\ell_{B}(g):=\min\\{n\in\mathbb{Z}_{\geq 0}\mid\exists b:\\{1,\dots,n\\}\to
B\text{ with }g=b(1)\cdots b(n)\\}$

(with the convention $\ell_{B}(1_{G})=0$). Clearly, a directed path
$1_{G}=g_{0},g_{1},\dots,g_{d}=g$ from $1_{G}$ to $g$ in $\mathrm{Cay}(G,B)$
is _geodesic_ if and only if $d=\ell_{B}(g)$. Therefore

(1) $\mathrm{diam}(\mathrm{Cay}(G,B))=\max\\{\ell_{B}(g)\mid g\in G\\},$

and

(2) $\mathsf{dcd}(G)=\max\\{\ell_{B}(g)\mid g\in G,\ B\subseteq G,\ \langle
B\rangle=G\\}.$

###  2.2. The monoid of product-one sequences

Denote by $(\mathcal{F}(G),\boldsymbol{\cdot})$ the free abelian monoid with
basis $G$. Its elements will be called _sequences over $G$_, because the
element

$S=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}\in\mathcal{F}(G)$

can be thought of as the sequence $g_{1},\dots,g_{\ell}$, where repetition is
allowed and the order of terms is disregarded. The number $\ell$ is called the
_length_ of $S$, denoted by $|S|$. Note that the product of
$g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}$ and
$h_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}h_{k}$ in the monoid
$\mathcal{F}(G)$ is
$g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}\boldsymbol{\cdot}h_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}h_{k}$
(concatenation of sequences). For
$S=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}\in\mathcal{F}(G)$ we
shall use the notation

$\mathsf{v}_{g}(S):=|\\{i\in\\{1,\dots,\ell\\}\mid g_{i}=g\\}|.$

For $g\in G$ and a positive integer $n$ we shall write $g^{[n]}$ for the
sequence $\underbrace{g\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g}_{n}$.
Define the _support_ of $S\in\mathcal{F}(G)$ as

$\mathrm{supp}(S):=\\{g\in G\mid\mathsf{v}_{g}(S)>0\\}.$

For $g\in\mathrm{supp}(S)$ denote by $S\boldsymbol{\cdot}g^{[-1]}$ the element
of $\mathcal{F}(G)$ defined by the equality

$S=g\boldsymbol{\cdot}(S\boldsymbol{\cdot}g^{[-1]}).$

The sequence $g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}$ is a
_product-one sequence_ if there exists a permutation $\pi\in S_{\ell}$ such
that $g_{\pi(1)}\cdots g_{\pi(\ell)}=1_{G}\in G$. Product-one sequences form a
submonoid $\mathcal{B}(G)$ of $\mathcal{F}(G)$. An element
$S\in\mathcal{B}(G)$ is called an _atom_ if it can not be written as
$S=T\boldsymbol{\cdot}U\in\mathcal{F}(G)$ with $T,U\in\mathcal{B}(G)$ and
$|T|>0$, $|U|>0$. Write $\mathcal{A}(G)$ for the set of atoms in
$\mathcal{B}(G)$. The maximal length of an atom in the monoid $\mathcal{B}(G)$
is denoted by $\mathsf{D}(G)$ and is called the _large Davenport constant of
$G$_. A sequence
$S=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}\in\mathcal{F}(G)$ is
_product-one free_ if $S$ can not be written as
$S=T\boldsymbol{\cdot}U\in\mathcal{F}(G)$ where $T\in\mathcal{B}(G)$ and
$|T|>0$. The maximal length of a product-one free sequence is denoted by
$\mathsf{d}(G)$ and is called the _small Davenport constant of $G$_. The small
and large Davenport constants are related by the following inequality:
$\mathsf{D}(G)\geq 1+\mathsf{d}(G)$.

##  3\. Abelian groups

When $G$ is abelian, we shall use the additive notation. A product-one
sequence $S=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}$ in this
case is rather called a _zero-sum sequence_. It means that
$g_{1}+\cdots+g_{\ell}=0\in G$. A product-one free sequence is called a zero-
sum free sequence in this case. So
$S=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{\ell}$ is _zero-sum free_
if there is no non-empty subset
$\\{i_{1},\dots,i_{k}\\}\subseteq\\{1,\dots,\ell\\}$ with
$g_{i_{1}}+\cdots+g_{i_{k}}=0\in G$. The small and large Davenport constants
of a finite abelian group $G$ are related by the equality
$\mathsf{D}(G)=\mathsf{d}(G)+1$.

######  Proposition 3.1.

For a finite abelian group $G$ we have $\mathsf{d}(G)\geq\mathsf{dcd}(G)$.

###### Proof.

Write $d:=\mathsf{dcd}(G)$. If $d=0$, then $G$ is the one-element group, and
both $\mathsf{d}(G)$ and $\mathsf{dcd}(G)$ are equal to zero. From now on we
assume that $d>0$. By (2) there exists an equality $g=g_{1}+\cdots+g_{d}$,
with $\\{g_{1},\dots,g_{d}\\}\subseteq B$, where $B$ is an irredundant
generating system of $G$, and $d=\ell_{B}(g)$. Then $g\neq 0$, and
$g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{d}$ is a zero-sum free
sequence, and so $\mathsf{d}(G)\geq d$. Indeed, otherwise after a possible
renumbering we have that for some $k<d$ we have $g_{k+1}+\cdots+g_{d}=0$,
implying that $g_{1}+\cdots+g_{k}=g$. Thus $\ell_{B}(g)\leq k<d$, contrary to
the assumption that $\ell_{B}(g)=d$. ∎

For an abelian group $G=C_{n_{1}}\oplus\cdots\oplus C_{n_{r}}$ with $2\leq
n_{1}\mid\dots\mid n_{r}$, set

$\mathsf{d}^{*}(G):=\sum_{i=1}^{r}(n_{i}-1).$

By [14, Lemma 4.1] (see also [12, Exercise 1.6]) we have

(3) $\mathsf{d}^{*}(G)\geq\mathsf{d}^{*}(H)+\mathsf{d}^{*}(G/H)\text{ for any
subgroup }H\text{ of }G.$

In particular,

(4) $\mathsf{d}^{*}(H)+|G:H|-1\leq\mathsf{d}^{*}(G)\quad\mbox{ when }G/H\mbox{
is cyclic}$

(see also [6, Lemma 3.2] for an alternative proof).

Note that $\mathsf{d}(G)\geq\mathsf{d}^{*}(G)$ for all $G$. Therefore
Proposition 3.1 follows from the sharper Theorem 3.2 below.

######  Theorem 3.2.

For any finite abelian group $G$ we have the equality

$\mathsf{dcd}(G)=\mathsf{d}^{*}(G).$

###### Proof.

For $i=1,\dots,r$ denote by $e_{i}$ the generator of the $i$th direct summand
$C_{n_{i}}$ of $G=C_{n_{1}}\oplus\cdots\oplus C_{n_{r}}$, where $2\leq
n_{1}\mid\dots\mid n_{r}$. Consider the generating system
$B:=\\{e_{1},\dots,e_{r}\\}$ of $G$. Then
$\ell_{B}(\sum_{i=1}^{r}(n_{i}-1)e_{i})=\sum_{i=1}^{r}(n_{i}-1)$, implying the
inequality $\mathsf{dcd}(G)\geq\sum_{i=1}^{r}(n_{i}-1)=\mathsf{d}^{*}(G)$.

We prove the reverse inequality $\mathsf{dcd}(G)\leq\mathsf{d}^{*}(G)$ by
induction on $|G|$. For $|G|=1,2,3$ the statement is obvious. Suppose $|G|\geq
4$, and take a generating system $B$ of $G$ such that
$d:=\mathsf{dcd}(G)=\mathrm{diam}(\mathrm{Cay}(G,B))$. For any subset
$C\subseteq B$ we obviously have
$\mathrm{diam}(\mathrm{Cay}(G,C))\geq\mathrm{diam}(\mathrm{Cay}(G,B))$,
therefore omitting redundant elements from $B$ we can make it irredundant.
Pick some $b\in B$. Then $B\setminus\\{b\\}$ generates a proper subgroup $H$
of $G$, therefore $\mathsf{dcd}(H)\leq\mathsf{d}^{*}(H)$ by the induction
hypothesis. Now take an element $g\in G$ with $\ell_{B}(g)=d$. Denote by
$\bar{g}$ the image of $g$ in the factor group $G/H$. The factor group $G/H$
is generated by the image $\bar{b}$ of $b$. So there exists a
$k\in\\{0,1,\dots,|G:H|-1\\}$ such that $\bar{g}=k\bar{b}$, or equivalently,
$g-kb\in H$. Now

$\ell_{B\setminus\\{b\\}}(g-kb)\leq\mathrm{diam}(\mathrm{Cay}(H,B\setminus\\{b\\}))\leq\mathsf{dcd}(H)\leq\mathsf{d}^{*}(H),$

hence

$d=\ell_{B}(g)\leq\ell_{B\setminus\\{b\\}}(g-kb)+k\leq\mathsf{d}^{*}(H)+k\leq\mathsf{d}^{*}(H)+|G:H|-1.$

Finally, by (4) we conclude $\mathsf{dcd}(G)=d\leq\mathsf{d}^{*}(G)$. ∎

######  Remark 3.3.

  * (i)

An easy short argument yielding an upper bound for $\mathsf{dcd}(G)$ in the
case when $G$ is the direct sum of prime order groups is given in [3, Lemma
5.2].

  * (ii)

The undirected Cayley diameter of a finite abelian group is determined in [22,
Lemma 3.3]. For $G=C_{n_{1}}\oplus\cdots\oplus C_{n_{r}}$ with $2\leq
n_{1}\mid\dots\mid n_{r}$, it is $\sum_{i=1}^{r}\lfloor n_{i}/2\rfloor$.

The equality $\mathsf{d}(G)=\mathsf{d}^{*}(G)$ is known to hold for abelian
groups of rank at most $2$, for abelian $p$-groups, and for several other
infinite classes of abelian groups, and this equality is conjectured to hold
for abelian groups of rank $3$ and for the groups of the form
$C_{n}\oplus\cdots\oplus C_{n}$. Therefore for a notable class of abelian
groups the inequality in the statement of Proposition 3.1 holds with equality.

######  Example 3.4.

Consider the group $G:=C_{3}\oplus C_{3}\oplus C_{3}\oplus C_{6}$ with
generators $e_{i}$ $(i=1,\dots,4)$ such that
$\mathrm{ord}(e_{1})=\mathrm{ord}(e_{2})=\mathrm{ord}(e_{3})=3$ and
$\mathrm{ord}(e_{4})=6$. Set $g_{1}:=-e_{1}+e_{2}+e_{3}+e_{4}$,
$g_{2}:=e_{1}-e_{2}+e_{3}+e_{4}$, $g_{3}:=e_{1}+e_{2}-e_{3}+e_{4}$,
$g_{4}:=-e_{1}+e_{2}+e_{3}-e_{4}$, $g_{5}=e_{1}+e_{2}+e_{3}+e_{4}$,
$g_{6}:=e_{2}-e_{3}-e_{4}$, $g_{7}:=-e_{2}+e_{3}-e_{4}$. It is shown (as a
special case of a more general statement) in [11] that the sequence
$g_{1}^{[2]}\boldsymbol{\cdot}g_{2}^{[2]}\boldsymbol{\cdot}g_{3}^{[2]}\boldsymbol{\cdot}g_{4}^{[2]}\boldsymbol{\cdot}g_{5}^{[2]}\boldsymbol{\cdot}g_{6}\boldsymbol{\cdot}g_{7}$
is zero-sum free. As this sequence has length $12$, we conclude that
$\mathsf{d}(G)\geq 12>11=\mathsf{d}^{*}(G)$. Any ordering of the elements in
this sequence gives a path of length $12$ from $0$ to
$g:=2(g_{1}+g_{2}+g_{3}+g_{4}+g_{5})+g_{6}+g_{7}$ in the Cayley graph
$\mathrm{Cay}(G,\\{g_{i}\mid i=1,\dots,7\\})$. Such a path can not be a
geodesic path from $0$ to $g$ in $\mathrm{Cay}(G,\\{g_{i}\mid i=1,\dots,7\\})$
by Theorem 3.2. Indeed, the equality

$2(g_{1}+g_{2}+g_{3}+g_{4}+g_{5})+g_{6}+g_{7}=4g_{1}+g_{2}+g_{3}+2g_{4},$

shows that there exists a path of length $4+1+1+2=8$ from $0$ to $g$ in
$\mathrm{Cay}(G,\\{g_{i}\mid i=1,\dots,7\\})$, hence $\ell_{\\{g_{i}\mid
i=1,\dots,7\\}}(g)\leq 8$.

##  4\. The relation between the Davenport constant and the directed Cayley
diameter

In view of Proposition 3.1 it is natural to pose the following question: does
the inequality $\mathsf{dcd}(G)\leq\mathsf{d}(G)$ hold for all non-abelian
finite groups $G$?

The computer calculations presented in Section 8 answer this question in the
negative, see for example the group with GAP-identifier $[16,3]$ in the table
in Section 8. What we know instead is that the directed Cayley diameter can be
bounded in terms of the large Davenport constant, since in [5, Proposition
5.6] the following inequality was pointed out:

######  Proposition 4.1.

We have $\mathsf{D}(G)>\mathsf{dcd}(G)$ for all finite groups $G$.

Proposition 4.1 follows from the following observation:

######  Lemma 4.2.

Suppose that $\ell_{\\{b_{1},\dots,b_{d}\\}}(b_{1}\cdots b_{d})=d$ for a
sequence
$b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\in\mathcal{F}(G)$. Then
setting $b_{d+1}:=(b_{1}\cdots b_{d})^{-1}$, we have that
$b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\boldsymbol{\cdot}b_{d+1}$
is an atom in $\mathcal{B}(G)$.

###### Proof.

Suppose for contradiction that
$b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\boldsymbol{\cdot}b_{d+1}=S\boldsymbol{\cdot}T$
for some $S,T\in\mathcal{B}(H)$ with $|S|>0$ and $|T|>0$. By symmetry we may
assume that $\mathsf{v}_{b_{d+1}}(T)\geq 1$, so
$T=c_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}c_{m}\boldsymbol{\cdot}b_{d+1}$
for some $c_{1},\dots,c_{m}\in\\{b_{1},\dots,b_{d}\\}$. Moreover, after a
possible renumbering we have $c_{1}\dots c_{m}b_{d+1}=1_{G}$. Thus
$c_{1}\cdots c_{m}=b_{d+1}^{-1}=b_{1}\cdots b_{d}$, hence
$\ell_{\\{b_{1},\dots,b_{d}\\}}(b_{d+1}^{-1})\leq\ell_{\\{c_{1},\dots,c_{m}\\}}(b_{d+1}^{-1})\leq
m<d$. This contradicts the assumption
$\ell_{\\{b_{1},\dots,b_{d}\\}}(b_{1}\cdots b_{d})=d$. ∎

Lemma 4.2 implies the following strengthening of Proposition 4.1:

######  Corollary 4.3.

For any finite group $G$ we have the inequality

(5) $\mathsf{D}(G)>\max\\{\mathsf{dcd}(H)\mid H\leq G\\}.$

######  Remark 4.4.

Corollary 4.3 is stronger than Proposition 4.1, since in some cases it happens
that $\mathsf{dcd}(H)>\mathsf{dcd}(G)$ for a subgroup $H$ of $G$ (see Remark
8.2).

##  5\. Geodesic atoms and their applications

The number on the right hand side of the inequality (5) can be characterized
in terms of the lengths of some special atoms in $\mathcal{B}(G)$.

######  Definition 5.1.

An element
$b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\boldsymbol{\cdot}b_{d+1}$
of $\mathcal{B}(G)$ is called a _geodesic atom_ if

  * (i)

$b_{1}\cdots b_{d}b_{d+1}=1_{G}$, and

  * (ii)

$\ell_{\\{b_{1},\dots,b_{d}\\}}(b_{d+1}^{-1})=d$.

Write $\mathcal{GA}(G)$ for the subset of geodesic atoms in $\mathcal{B}(G)$.

######  Remark 5.2.

The terminology “geodesic atom” is motivated by the following:

  * (i)

Conditions (i) and (ii) in Definition 5.1 are equivalent to the condition that
$1_{G},b_{1},b_{1}b_{2},\dots,b_{1}\cdots b_{d}$ is a geodesic path from
$1_{G}$ to $b_{1}\cdots b_{d}$ in the directed Cayley graph
$\mathrm{Cay}(\langle b_{1},\dots,b_{d}\rangle,\\{b_{1},\dots,b_{d}\\})$.

  * (ii)

By Lemma 4.2 a geodesic atom in $\mathcal{B}(G)$ is an atom in
$\mathcal{B}(G)$, so

(6) $\mathcal{GA}(G)\subseteq\mathcal{A}(G).$

######  Example 5.3.

Keeping the notation of Example 3.4, set
$g_{8}:=-(2g_{1}+2g_{2}+2g_{3}+2g_{4}+2g_{5}+g_{6}+g_{7})$. The sequence
$g_{1}^{[2]}\boldsymbol{\cdot}g_{2}^{[2]}\boldsymbol{\cdot}g_{3}^{[2]}\boldsymbol{\cdot}g_{4}^{[2]}\boldsymbol{\cdot}g_{5}^{[2]}\boldsymbol{\cdot}g_{6}\boldsymbol{\cdot}g_{7}\boldsymbol{\cdot}g_{8}$
is an atom in $\mathcal{B}(C_{3}\oplus C_{3}\oplus C_{3}\oplus C_{6})$, but
our considerations in Section 3 show that it is not a geodesic atom.

In analogy with the large Davenport constant, we define the following
quantity:

######  Definition 5.4.

  * (i)

The _geodesic large Davenport constant of_ $G$ is

$\mathsf{GD}(G):=\max\\{|S|\mid S\in\mathcal{GA}(G)\\}.$

  * (ii)

Set

$\mathsf{dcd}^{*}(G):=\max\\{|S|\colon S\in\mathcal{GA}(G),\
\langle\operatorname{supp}(S)\rangle=G\\}-1.$

As an immediate consequence of (2) and Definition 5.1 we get the following:

######  Lemma 5.5.

We have

  * (i)

$\mathsf{dcd}(G)\geq\mathsf{dcd}^{*}(G)$.

  * (ii)

$\max\\{\mathsf{dcd}(H)\mid H\leq G\\}=\mathsf{GD}(G)-1$.

######  Corollary 5.6.

For any finite group $G$ we have

$\mathsf{dcd}^{*}(G)\leq\mathsf{dcd}(G)\leq\mathsf{GD}(G)-1\leq\mathsf{D}(G)-1.$

###### Proof.

The first two inequalities follow from Lemma 5.5, the third inequality follows
from (6). ∎

Lemma 5.5 (ii) implies that $\mathsf{dcd}(G)$ and $\mathsf{GD}(G)$ are tightly
linked for certain groups as follows:

######  Corollary 5.7.

The following conditions are equivalent for a finite group $G$:

  * (i)

$\mathsf{dcd}(G)\geq\mathsf{dcd}(H)$ for all subgroups $H$ of $G$.

  * (ii)

$\mathsf{dcd}(G)=\mathsf{GD}(G)-1$.

Moreover, if $\mathsf{dcd}^{*}(G)=\mathsf{GD}(G)-1$, then (i) and (ii) hold.

Generating systems of $G$ can also be read off from $\mathcal{GA}(G)$ (and
hence from $\mathcal{A}(G)$):

######  Proposition 5.8.

A subset $B$ generates the finite group $G$ if and only if for all $g\in G$
there exists an $S\in\mathcal{GA}(G)$ with $g\in\mathrm{supp}(S)$ and
$\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})\subseteq B$.

###### Proof.

Suppose that $B$ generates $G$. Then for each $g\in G$ there exist
$b_{1},\dots,b_{n}\in B$ with $g^{-1}=b_{1}\cdots b_{n}$, and
$n=\ell_{B}(g^{-1})$. Now
$S:=b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{n}\boldsymbol{\cdot}g$
belongs to $\mathcal{GA}(G)$ and $S$ satisfies $g\in\mathrm{supp}(S)$ and
$\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})\subseteq B$.

Conversely, let $B$ be a subset of $G$ such that for all $g\in G$ there exists
an $S\in\mathcal{GA}(G)$ with $g\in\mathrm{supp}(S)$ and
$\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})\subseteq B$. Then in particular,
for all $g\in G$ there exist elements $b_{1},\dots,b_{n}\in B$ with
$b_{1}\cdots b_{n}g=1_{G}$, i.e. $g^{-1}=b_{1}\cdots b_{n}$. That is, $B$
generates $G$. ∎

Given the set $\mathcal{GA}(G)$ and a fixed generating system $B$ of $G$, it
is in principle an easy matter to determine
$\mathrm{diam}(\mathrm{Cay}(G,B))$. Select the following subsets of
$\mathcal{GA}(G)$:

$\mathcal{GA}(G,B,0):=\\{S\in\mathcal{GA}(G)\mid\mathrm{supp}(S)\subseteq
B\\}$

For $g\in G\setminus B$ set

$\mathcal{GA}(G,B,g,1):=\\{S\in\mathcal{GA}(G)\mid\mathsf{v}_{g}(S)=1,\
\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})\subseteq B\\},$

and set

$\mathcal{GA}(G,B,g):=\\{S\in\mathcal{GA}(G,B,g,1)\mid\forall
T\in\mathcal{GA}(G,B,g,1):|T|\geq|S|\\}.$

Finally,

$\mathcal{GA}(G,B):=\mathcal{GA}(G,B,0)\cup\bigcup_{g\in G\setminus
B}\mathcal{GA}(G,B,g).$

######  Proposition 5.9.

We have the equality

(7) $\mathrm{diam}(\mathrm{Cay}(G,B))=\max\\{|S|\mid
S\in\mathcal{GA}(G,B)\\}-1.$

###### Proof.

Set $d:=\mathrm{diam}(\mathrm{Cay}(G,B))$ and write $m$ for the number on the
right hand side of (7). First we show that $m\geq d$. By definition of $d$
(see (1)) there exists an element $h:=b_{1}\cdots b_{d}$, where
$b_{1},\dots,b_{d}\in B$ and $\ell_{B}(h)=d$. Then for $g:=h^{-1}$ we have
that $b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\boldsymbol{\cdot}g$
is a geodesic atom, and it belongs to $\mathcal{GA}(G,B,0)$ if $g\in B$, and
it belongs to $\mathcal{GA}(G,B,g)$ if $g\in G\setminus B$. This shows that
$m\geq d$.

By definition of $m$ there exists a geodesic atom
$S=b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{m}\boldsymbol{\cdot}g\in\mathcal{GA}(G,B)$,
so $b_{1},\dots,b_{m}\in B$ and $\ell_{B}(g^{-1})=m$, showing that the reverse
inequality $d\geq m$ holds. ∎

##  6\. Characterization of geodesic atoms

The results of Section 5 show that if we are given the set $\mathcal{GA}(G)$
of geodesic atoms for $G$, then we can read off generating systems of $G$,
diameters of Cayley digraphs of $G$, or the directed Cayley diameter of $G$.
This motivates the search for a characterizetion of the geodesic atoms in the
set of all atoms $\mathcal{A}(G)$.

######  Lemma 6.1.

Suppose that $S\in\mathcal{B}(G)$ is a geodesic atom. Then there exists a
$g\in\mathrm{supp}(S)$ such that the following holds: for all
$T\in\mathcal{B}(G)$ with $g\in\mathrm{supp}(T)$ and
$\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})$,
we have $|T|\geq|S|$.

###### Proof.

Since $S\in\mathcal{GA}(G)$, we can write
$S=b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\boldsymbol{\cdot}b_{d+1}$,
where $b_{1}\cdots b_{d}b_{d+1}=1_{G}$, and setting
$B:=\\{b_{1},\dots,b_{d}\\}$, we have $\ell_{B}(b_{d+1}^{-1})=d$. Now for
$g:=b_{d+1}$ we have $\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})=B$. Take any
$T\in\mathcal{B}(G)$ with $g\in\mathrm{supp}(T)$ and
$\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq B$. Then
$T=c_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}c_{k}\boldsymbol{\cdot}g$
with $c_{i}\in B$. With an appropriate numbering of the $c_{i}$, we have
$c_{1}\cdots c_{k}g=1_{G}$, so $c_{1}\cdots c_{k}=g^{-1}=b_{d+1}^{-1}$. It
follows that $k\geq\ell_{B}(b_{d+1}^{-1})=d$, implying in turn that
$|T|=k+1\geq d+1=|S|$. This finishes the proof. ∎

######  Lemma 6.2.

Suppose that $S\in\mathcal{B}(G)$ is not a geodesic atom. Then for each
$g\in\mathrm{supp}(S)$ there exists a $T\in\mathcal{GA}(G)$ such that the
following hold:

  * (i)

$|T|<|S|$;

  * (ii)

$g\in\mathrm{supp}(T)$;

  * (iii)

$\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})$.

###### Proof.

Assume $S\in\mathcal{B}(G)\setminus\mathcal{GA}(G)$. Pick an arbitrary $g$
from $\mathrm{supp}(S)$. Then we can write
$S=b_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}b_{d}\boldsymbol{\cdot}g$,
with $g^{-1}=b_{1}\cdots b_{d}$. Since $S\notin\mathcal{GA}(G)$, necessarily
we have $\ell_{B}(g^{-1})<d$, where $B=\\{b_{1},\dots,b_{d}\\}$. It follows
that setting $k:=\ell_{B}(g^{-1})$, there exist elements $c_{1},\dots,c_{k}\in
B$ such that $g^{-1}=c_{1}\cdots c_{k}$. Set
$T:=c_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}c_{k}\boldsymbol{\cdot}g$.
Then $T\in\mathcal{GA}(G)$ by Definition 5.1. Moreover, $|T|=k+1<d+1=|S|$,
$g\in\mathrm{supp}(T)$, and
$\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq
B=\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})$. ∎

Lemma 6.1 and Lemma 6.2 have the following immediate consequence:

######  Corollary 6.3.

The following conditions are equivalent for an element $S\in\mathcal{B}(G)$:

  * (i)

$S$ is not a geodesic atom.

  * (ii)

For all $g\in\mathrm{supp}(S)$ there exists a $T\in\mathcal{GA}(G)$ with
$|T|<|S|$, such that $g\in\mathrm{supp}(T)$ and
$\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})$.

  * (iii)

For all $g\in\mathrm{supp}(S)$ there exists a $T\in\mathcal{A}(G)$ with
$|T|<|S|$, such that $g\in\mathrm{supp}(T)$ and
$\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq\mathrm{supp}(S\boldsymbol{\cdot}g^{[-1]})$.

######  Remark 6.4.

An algorithm to compute the set $\mathcal{A}(G)$ of atoms in $\mathcal{B}(G)$
is given in [5, Section 6]. Now Corollary 6.3 furnishes a simple algorithm to
build up the set $\mathcal{GA}(G)$ from $\mathcal{A}(G)$. Namely, suppose we
found all elements in $\mathcal{GA}(G)$ with length at most $t$. Then we have
to go through all the atoms in $\mathcal{A}(G)$ of length $t+1$, and by
Corollary 6.3 (or by Lemma 6.2) it is easy to select those which belong to
$\mathcal{GA}(G)$. In Section 7 we shall present an algorithm which computes
directly $\mathcal{GA}(G)$, without a prior computation of $\mathcal{A}(G)$.

##  7\. Computing $\mathsf{GD}(G)$ and $\mathsf{dcd}^{*}(G)$

Recall the following definition from [5]:

######  Definition 7.1.

Let $S,T\in\mathcal{F}(G)$. The sequence $T$ is called _a splitting_ of the
sequence $S$ if
$T=S\boldsymbol{\cdot}g^{[-1]}\boldsymbol{\cdot}x\boldsymbol{\cdot}y$ for some
$g\in\operatorname{supp}(S)$ and $x,y\in G$ such that $xy=g$. We denote by
$S\prec T$ the fact that $T$ is a splitting of $S$. For any subset
$\mathcal{S}\subset\mathcal{F}(G)$ we set
$\gamma(\mathcal{S}):=\\{T\in\mathcal{F}(G)\mid S\prec T\text{ for some
}S\in\mathcal{S}\\}$.

Let us denote simply by $\mathcal{GA}_{k}$ the set of all geodesic atoms of
length $k\geq 0$ in $\mathcal{GA}(G)$ and by $\mathcal{GR}_{k}$ a set of
$\operatorname{Aut}(G)$-orbit representatives of $\mathcal{GA}_{k}$ (here
$\operatorname{Aut}(G)$ stands for the automorphism group of $G$). The
possibility to build up $\mathcal{GA}_{k}$ recursively is based on the
following lemma:

######  Lemma 7.2.

For any $k\geq 1$ we have
$\mathcal{GA}_{k+1}\subseteq\gamma(\mathcal{GA}_{k})$.

###### Proof.

All product-one sequences of length one or two (except the sequence
$1_{G}\boldsymbol{\cdot}1_{G}$) are geodesic atoms. Assume that $k\geq 2$, and
take an arbitrary $S\in\mathcal{GA}_{k+1}$. Then

$S=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{k}\boldsymbol{\cdot}g_{k+1},\mbox{
where }g_{1}\dots g_{k}g_{k+1}=1_{G}\mbox{ and
}\ell_{\\{g_{1},\dots,g_{k}\\}}(g_{k+1}^{-1})=k.$

Set

$T:=g_{1}\boldsymbol{\cdot}\dots\boldsymbol{\cdot}g_{k-1}\boldsymbol{\cdot}h\quad\mbox{where}\quad
h:=g_{k}g_{k+1}.$

Then obviously we have $g_{1}\dots g_{k-1}h=1_{G}$. We claim that $T$ is a
geodesic atom. Suppose for contradiction that

$h^{-1}=c_{1}\dots c_{m}\mbox{ for some }m<k-1\mbox{ and
}c_{1},\dots,c_{m}\in\\{g_{1},\dots,g_{k-1}\\}.$

Taking into account $h^{-1}=g_{k+1}^{-1}g_{k}^{-1}$ we conclude

$g_{k+1}^{-1}=c_{1}\dots c_{m}g_{k},\quad\mbox{where}\quad
c_{1},\dots,c_{m},g_{k}\in\\{g_{1},\dots,g_{k}\\}.$

Thus

$\ell_{\\{g_{1},\dots,g_{k}\\}}(g_{k+1}^{-1})\leq m+1<k,$

a contradiction. Therefore $T\in\mathcal{GA}_{k}$ and $T\prec S$, hence
$S\in\gamma(\mathcal{GA}_{k})$. Since $S$ above was an arbitrary element of
$\mathcal{GA}_{k+1}$, the statement is proved. ∎

######  Remark 7.3.

Observe that by this lemma if $\mathcal{GA}_{k}=\emptyset$ for some $k\geq 1$
then $\mathcal{GA}_{l}=\emptyset$ for every $l>k$. So the smallest $k$ such
that $\mathcal{GA}_{k}=\emptyset$ equals $\mathsf{GD}(G)+1$.

######  Lemma 7.4.

For any $k\geq 1$ let $\mathcal{GR}_{k}$ be a set of
$\operatorname{Aut}(G)$-orbit representatives in $\mathcal{GA}_{k}$. Then
there is a system of $\operatorname{Aut}(G)$-orbit representatives
$\mathcal{GR}_{k+1}$ in $\mathcal{GA}_{k+1}$ such that
$\mathcal{GR}_{k+1}\subseteq\gamma(\mathcal{GR}_{k})$.

###### Proof.

Observe that the relation $\prec$ is compatible with the
$\operatorname{Aut}(G)$-action in the sense that $S\prec T$ holds if and only
if $\alpha(S)\prec\alpha(T)$ for some $\alpha\in\operatorname{Aut}(G)$ if and
only if $\alpha(S)\prec\alpha(T)$ for each $\alpha\in\operatorname{Aut}(G)$.

Now we prove that for any sequence $T\in\mathcal{GA}_{k+1}$ its
$\operatorname{Aut}(G)$-orbit $\operatorname{Orb}(T)$ has a non-empty
intersection with $\gamma(\mathcal{GR}_{k})$. By Lemma 7.2 there exists a
sequence $S\prec T$ so that $S\in\mathcal{GA}_{k}$. As
$\mathcal{GR}_{k}\subseteq\mathcal{GA}_{k}$ is a complete set of
$\operatorname{Aut}(G)$-orbit representatives, there is some
$R\in\mathcal{GR}_{k}$ such that $R=\alpha(S)$ for some
$\alpha\in\operatorname{Aut}(G)$. Hence by the compatibility of $\prec$ it
follows that $R=\alpha(S)\prec\alpha(T)$. As a result
$\alpha(T)\in\operatorname{Orb}(T)\cap\gamma(\mathcal{GR}_{k})$. ∎

Given an arbitrary finite group $G$, Algorithm 1 below computes the geodesic
large Davenport constant $\mathsf{GD}(G)$. The set of all geodesic atoms of
$G$ may be obtained as $\mathcal{GA}(G)=\bigcup_{k}\mathcal{GA}_{k}$ after the
repeat-until loop has terminated (right after line 1).

The algorithm begins with initializing the sets $\mathcal{GA}_{1}$ and
$\mathcal{GR}_{1}$ with the sequence consisting only of the identity element
of the group. In line 1 the variable $S^{\prime}$ will take successively all
the values from the set $\gamma(\mathcal{GR}_{k-1})$, therefore it is
guaranteed by Lemma 7.4 that for $k>1$ all the possible candidate sequences
will be tested for being geodesic atoms and the set $\mathcal{GA}_{k}$ is
built correctly. In line 1 the variable $x$ is prevented from taking the
values $1_{G}$ or $g$, since this would bring in the identity element in the
support of $S^{\prime}$.

The algorithm first checks whether a candidate sequence $S^{\prime}$ is among
the already computed geodesic atoms of length $k$ via the condition
$S^{\prime}\notin\mathcal{GA}_{k}$ from line 1, ensuring that $S^{\prime}$ is
a new geodesic atom (in the sense that it is not similar to any of the already
collected geodesic atoms of length $k$). Checking whether a candidate sequence
$S^{\prime}$ is a geodesic atom is performed by the second condition in line
1, based on Corollary 6.3 (ii). The predicate
$\mathbf{NG}(S^{\prime},\cup_{i=1}^{k-1}\mathcal{GA}_{i},k)$ in line 1 is true
if and only if the product-one sequence $S^{\prime}$ of length $k$ is not a
geodesic atom and may be given in the following way:

$\displaystyle\mathbf{NG}(S^{\prime},\cup_{i=1}^{k-1}\mathcal{GA}_{i},k)\iff$
$\displaystyle\forall g\in\mathrm{supp}(S^{\prime}),\exists
T\in\cup_{i=1}^{k-1}\mathcal{GA}_{i}\text{ such that}$ $\displaystyle
g\in\mathrm{supp}(T)\text{ and
}\mathrm{supp}(T\boldsymbol{\cdot}g^{[-1]})\subseteq\mathrm{supp}(S^{\prime}\boldsymbol{\cdot}g^{[-1]}).$

In line 1 the newly found geodesic atom $S^{\prime}$ gets added to the set
$\mathcal{GR}_{k}$ of representatives and the set $\mathcal{GA}_{k}$ of
geodesic atoms is completed with the $\operatorname{Aut}(G)$ orbit of
$S^{\prime}$, denoted by $\operatorname{Orb}(S^{\prime})$ (in line 1). In the
last step, the value $k-1$ is returned as the geodesic large Davenport
constant of $G$, as explained in Remark 7.3.

1

2

3

input : $G$

output : $\mathsf{GD}(G)$

4

5

6$\mathcal{GA}_{1}\leftarrow\\{1_{G}\\}$;

7 $\mathcal{GR}_{1}\leftarrow\\{1_{G}\\}$;

8 $k\leftarrow 1$;

9

10repeat

11 $k\leftarrow k+1$;

12 $\mathcal{GA}_{k}\leftarrow\emptyset$;

13 $\mathcal{GR}_{k}\leftarrow\emptyset$;

14 forall  _$S\in\mathcal{GR}_{k-1}$_ do

15 forall  _$g\in\operatorname{supp}(S)$_ do

16 forall  _$x\in G\setminus\\{1_{G},g\\}$_ do

17 $S^{\prime}\leftarrow
S\boldsymbol{\cdot}g^{[-1]}\boldsymbol{\cdot}x\boldsymbol{\cdot}(x^{-1}g)$;

18 if  _$S^{\prime}\notin\mathcal{GA}_{k}$ and not
$\mathbf{NG}(S^{\prime},\cup_{i=1}^{k-1}\mathcal{GA}_{i},k)$_ then

19 $\mathcal{GR}_{k}\leftarrow\mathcal{GR}_{k}\cup\\{S^{\prime}\\}$;

20
$\mathcal{GA}_{k}\leftarrow\mathcal{GA}_{k}\cup\operatorname{Orb}(S^{\prime})$;

21

22 end if

23

24 end forall

25

26 end forall

27

28 end forall

29

30until  _$\mathcal{GA}_{k}=\emptyset$_ ;

31

32return $k-1$;

33

Algorithm 1 GeodesicLargeDavenport($G$)

One could perform a slight modification of Algorithm 1 in Line 1 to return
also the sequence of sets $(\mathcal{GR}_{i})_{1\leq i\leq\mathsf{GD}(G)}$.
With this information, we can devise Algorithm 2, which, given an arbitrary
finite group $G$, computes $\mathsf{dcd}^{*}(G)$, yielding a lower bound for
the directed Cayley diameter of $G$ by Lemma 5.5 (i):

1

2

3

input : $G$

output : $\mathsf{dcd}^{*}(G)$

4

5

6$\mathsf{GD}(G),\ (\mathcal{GR}_{i})_{1\leq
i\leq\mathsf{GD}(G)}\leftarrow\mathrm{GeodesicLargeDavenport}(G)$;

7 $k\leftarrow\mathsf{GD}(G)+1$;

8 $dcdstar\\_found\leftarrow\textsf{false}$;

9 repeat

10 $k\leftarrow k-1$;

11 forall  _$S\in\mathcal{GR}_{k}$_ do

12 if  _$G=\langle\operatorname{supp}(S)\rangle$_ then

13 $dcdstar\\_found\leftarrow\textsf{true}$;

14 break ;

15

16 end if

17

18 end forall

19

20until  _$dcdstar\\_found$_ ;

21return $k-1$;

22

Algorithm 2 DirectedCayleyDiameterLowerBound($G$)

The algorithm begins by calling (the slightly modified)
$\mathrm{GeodesicLargeDavenport}(G)$, which in turn provides the value for
$\mathsf{GD}(G)$ and all the sets $\mathcal{GR}_{k}$ for $1\leq
k\leq\mathsf{GD}(G)$ (in line 2). Then a simple implementation of Definition
5.4 (ii) follows.

######  Remark 7.5.

The condition in line 2 can be implemented in GAP (see [15]) using the
$\mathsf{Group(gens)}$ command, where $\mathsf{gens}$ is a list of generators.
Another possibility would be to use the sets $\mathcal{GA}_{k}$ to decide
whether $\operatorname{supp}(S)$ generates $G$ by Proposition 5.8.

##  8\. Results of computer calculations

In the table below we give $\mathsf{dcd}^{*}(G)$, $\mathsf{d}(G)$,
$\mathsf{GD}(G)$, $\mathsf{D}(G)$ for all non-abelian groups of order at most
$39$, and in several cases, we give also the value for $\mathsf{dcd}(G)$, when
it happens that $\mathsf{dcd}^{*}(G)=\mathsf{dcd}(G)=\mathsf{GD}(G)-1$ (see
Corollary 5.7). The first column contains the identifier of $G$ in the Small
Groups Library of GAP (see [15]); that is, the groups from the table below can
be obtained using the GAP command $\mathsf{SmallGroup(id)}$. The second column
contains the “structure description” string of the group, as returned by the
$\mathsf{StructureDescription(G)}$ command.

id | Struct. desc. | $\mathsf{dcd}^{*}(G)$ | $\mathsf{dcd}(G)$ | $\mathsf{d}(G)$ | $\mathsf{GD}(G)$ | $\mathsf{D}(G)$
---|---|---|---|---|---|---
[6, 1] | S3 | 3 | 3 | 3 | 4 | 6
[8, 3] | D8 | 4 | 4 | 4 | 5 | 6
[8, 4] | Q8 | 3 | 3 | 4 | 4 | 6
[10, 1] | D10 | 5 | 5 | 5 | 6 | 10
[12, 1] | C3 : C4 | 5 | 5 | 6 | 6 | 9
[12, 3] | A4 | 4 | 4 | 4 | 5 | 7
[12, 4] | D12 | 6 | 6 | 6 | 7 | 9
[14, 1] | D14 | 7 | 7 | 7 | 8 | 14
[16, 3] | (C4 x C2) : C2 | 6 | 6 | 5 | 7 | 7
[16, 4] | C4 : C4 | 6 | 6 | 6 | 7 | 8
[16, 6] | C8 : C2 | 6 | - | 8 | 8 | 10
[16, 7] | D16 | 8 | 8 | 8 | 9 | 12
[16, 8] | QD16 | 5 | - | 8 | 8 | 12
[16, 9] | Q16 | 5 | - | 8 | 8 | 12
[16, 11] | C2 x D8 | 5 | 5 | 5 | 6 | 7
[16, 12] | C2 x Q8 | 4 | 4 | 5 | 5 | 7
[16, 13] | (C4 x C2) : C2 | 4 | 4 | 5 | 5 | 7
[18, 1] | D18 | 9 | 9 | 9 | 10 | 18
[18, 3] | C3 x S3 | 7 | 7 | 7 | 8 | 10
[18, 4] | (C3 x C3) : C2 | 4 | 4 | 5 | 5 | 10
[20, 1] | C5 : C4 | 7 | - | 10 | 10 | 15
[20, 3] | C5 : C4 | 6 | 6 | 7 | 7 | 10
[20, 4] | D20 | 10 | 10 | 10 | 11 | 15
[21, 1] | C7 : C3 | 5 | - | 8 | 7 | 14
[22, 1] | D22 | 11 | 11 | 11 | 12 | 22
[24, 1] | C3 : C8 | 9 | - | 12 | 12 | 15
[24, 3] | SL(2,3) | 6 | 6 | 7 | 7 | 13
[24, 4] | C3 : Q8 | 7 | - | 12 | 12 | 18
[24, 5] | C4 x S3 | 8 | - | 12 | 12 | 15
[24, 6] | D24 | 12 | 12 | 12 | 13 | 18
[24, 7] | C2 x (C3 : C4) | 8 | 8 | 8 | 9 | 11
[24, 8] | (C6 x C2) : C2 | 7 | 7 | 7 | 8 | 14
[24, 10] | C3 x D8 | 8 | - | 12 | 12 | 14
[24, 11] | C3 x Q8 | 8 | - | 12 | 12 | 14
[24, 12] | S4 | 7 | 7 | 6 | 8 | 12
[24, 13] | C2 x A4 | 7 | 7 | 7 | 8 | 10
[24, 14] | C2 x C2 x S3 | 7 | 7 | 7 | 8 | 10
[26, 1] | D26 | 13 | 13 | 13 | 14 | 26
[27, 3] | (C3 x C3) : C3 | 6 | 6 | 6 | 7 | 8
[27, 4] | C9 : C3 | 6 | - | 10 | 9 | 12
[28, 1] | C7 : C4 | 9 | - | 14 | 14 | 21
[28, 3] | D28 | 14 | 14 | 14 | 15 | 21
[30, 1] | C5 x S3 | 11 | - | 15 | 15 | 18
[30, 2] | C3 x D10 | 9 | - | 15 | 15 | 20
[30, 3] | D30 | 15 | 15 | 15 | 16 | 30
[32, 2] | (C4 x C2) : C4 | 8 | 8 | 7 | 9 | 9
[32, 4] | C8 : C4 | 7 | - | 10 | 9 | 12
[32, 5] | (C8 x C2) : C2 | 10 | 10 | 9 | 11 | 11
[32, 6] | (C2 x C2 x C2) : C4 | 8 | 8 | 7 | 9 | 10
[32, 7] | (C8 : C2) : C2 | 7 | 7 | 9 | 8 | 12
[32, 8] | (C2 x C2) . (C4 x C2) | 7 | 7 | 9 | 8 | 12
[32, 9] | (C8 x C2) : C2 | 10 | 10 | 9 | 11 | 13
[32, 10] | Q8 : C4 | 7 | - | 9 | 9 | 13
[32, 11] | (C4 x C4) : C2 | 8 | 8 | 9 | 9 | 14
[32, 12] | C4 : C8 | 10 | 10 | 10 | 11 | 12
[32, 13] | C8 : C4 | 7 | - | 10 | 9 | 14
[32, 14] | C8 : C4 | 10 | 10 | 10 | 11 | 14
[32, 15] | C4 . D8 = C4 . (C4 x C2) | 8 | 8 | 10 | 9 | 14
[32, 17] | C16 : C2 | 10 | - | 16 | 16 | 18
[32, 18] | D32 | 16 | 16 | 16 | 17 | 24
[32, 19] | QD32 | 9 | - | 16 | 16 | 24
[32, 20] | Q32 | 9 | - | 16 | 16 | 24
[32, 22] | C2 x ((C4 x C2) : C2) | 7 | 7 | 6 | 8 | 8
[32, 23] | C2 x (C4 : C4) | 7 | 7 | 7 | 8 | 9
[32, 24] | (C4 x C4) : C2 | 6 | 6 | 7 | 7 | 9
[32, 25] | C4 x D8 | 7 | 7 | 7 | 8 | 9
[32, 26] | C4 x Q8 | 6 | 6 | 7 | 7 | 9
[32, 27] | (C2 x C2 x C2 x C2) : C2 | 6 | 6 | 6 | 7 | 9
[32, 28] | (C4 x C2 x C2) : C2 | 6 | 6 | 7 | 7 | 10
[32, 29] | (C2 x Q8) : C2 | 6 | 6 | 7 | 7 | 10
[32, 30] | (C4 x C2 x C2) : C2 | 6 | 6 | 7 | 7 | 10
[32, 31] | (C4 x C4) : C2 | 6 | 6 | 7 | 7 | 10
[32, 32] | (C2 x C2) . (C2 x C2 x C2) | 5 | - | 7 | 7 | 10
[32, 33] | (C4 x C4) : C2 | 5 | - | 7 | 7 | 10
[32, 34] | (C4 x C4) : C2 | 6 | 6 | 7 | 7 | 10
[32, 35] | C4 : Q8 | 6 | 6 | 7 | 7 | 10
[32, 37] | C2 x (C8 : C2) | 7 | - | 9 | 9 | 11
[32, 38] | (C8 x C2) : C2 | 7 | - | 9 | 9 | 11
[32, 39] | C2 x D16 | 9 | 9 | 9 | 10 | 13
[32, 40] | C2 x QD16 | 6 | - | 9 | 9 | 13
[32, 41] | C2 x Q16 | 6 | - | 9 | 9 | 13
[32, 42] | (C8 x C2) : C2 | 6 | - | 9 | 9 | 13
[32, 43] | C8 : (C2 x C2) | 6 | - | 9 | 9 | 12
[32, 44] | (C2 x Q8) : C2 | 5 | - | 9 | 8 | 12
[32, 46] | C2 x C2 x D8 | 6 | 6 | 6 | 7 | 8
[32, 47] | C2 x C2 x Q8 | 5 | 5 | 6 | 6 | 8
[32, 48] | C2 x ((C4 x C2) : C2) | 5 | 5 | 6 | 6 | 8
[32, 49] | (C2 x C2 x C2) : (C2 x C2) | 4 | - | 6 | 6 | 8
[32, 50] | (C2 x Q8) : C2 | 4 | 4 | 6 | 5 | 8
[34, 1] | D34 | 17 | 17 | 17 | 18 | 34
[36, 1] | C9 : C4 | 11 | - | 18 | 18 | 27
[36, 3] | (C2 x C2) : C9 | 10 | 10 | 10 | 11 | 13
[36, 4] | D36 | 18 | 18 | 18 | 19 | 27
[36, 6] | C3 x (C3 : C4) | 13 | 13 | 13 | 14 | 16
[36, 7] | (C3 x C3) : C4 | 6 | - | 8 | 8 | 13
[36, 9] | (C3 x C3) : C4 | 8 | 8 | 7 | 9 | 12
[36, 10] | S3 x S3 | 8 | 8 | 8 | 9 | 14
[36, 11] | C3 x A4 | 8 | 8 | 8 | 9 | 11
[36, 12] | C6 x S3 | 10 | 10 | 10 | 11 | 13
[36, 13] | C2 x ((C3 x C3) : C2) | 7 | 7 | 8 | 8 | 13
[38, 1] | D38 | 19 | 19 | 19 | 20 | 38
[39, 1] | C13 : C3 | 7 | - | 14 | 13 | 26

######  Example 8.1.

The table above shows that for $3\leq n\leq 19$ the dihedral group $D_{2n}$
has directed Cayley diameter $n$. It is quite easy to see that
$\mathsf{dcd}(D_{2n})\geq n$ for all $n\geq 3$. Indeed, $D_{2n}$ has the
presentation with generators and relations as $D_{2n}=\langle a,b\mid
a^{2}=b^{2}=(ab)^{n}=1\rangle$. Now when $n$ is even, then
$\ell_{\\{a,b\\}}((ab)^{n/2})=n$, and when $n$ is odd, then
$\ell_{\\{a,b\\}}((ab)^{(n-1)/2}a)=n$.

######  Remark 8.2.

The quantities $\mathsf{D}(G)$, $\mathsf{d}(G)$, $\mathsf{GD}(G)$ are
obviously monotone in the sense that for any subgroup $H$ of $G$ we have
$\mathsf{D}(H)\leq\mathsf{D}(G)$, $\mathsf{d}(H)\leq\mathsf{d}(G)$,
$\mathsf{GD}(H)\leq\mathsf{GD}(G)$. The quantity $\mathsf{dcd}(G)$ is also
monotone for abelian groups $G$ by (3) and Theorem 3.2. On the other hand, the
quantity $\mathsf{dcd}(G)$ is not monotone for all finite groups $G$. This
follows from some known cases of Babai’s Conjecture from [2] mentioned in the
Introduction, because a non-abelian simple group may contain a “large” cyclic
subgroup, having “large” directed Cayley diameter.

######  Remark 8.3.

The computer calculations were performed on a server with 3.6 processors and 1
of RAM memory. The results for the groups of order at most $32$ have been
obtained in $30$ hours of computing time. The results for the remaining groups
(up to order $39$) have been obtained in about two weeks’ time.

##  9\. Some known results on the Davenport constants

The small Davenport constant for non-abelian groups was introduced in [19],
where it was shown that the inequality $\mathsf{d}(G)\leq\frac{1}{2}|G|$ holds
for any non-cyclic group $G$, with equality when $G$ has an index $2$
subgroup. The large Davenport constant was introduced in [9], where it was
proved that if $G$ has an index $2$ subgroup, then
$\mathsf{D}(G)=\mathsf{d}(G)+|G^{\prime}|$; here $|G^{\prime}|$ stands for the
commutator subgroup of $G$. Bounds on $\mathsf{d}(G)$ in terms of $|G|$ and
the smallest prime divisor $p$ of $|G|$ were given in a sequence of papers
(see [13], [8]). To the best of our knowledge, the strongest published bound
of this type is given in [21], where it is proved that
$\mathsf{d}(G)\leq|G|/p+p-2$.

## References

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