# Observation of Alfvén Wave Reflection in the Solar Chromosphere:
Ponderomotive Force and First Ionization Potential Effect

Mariarita Murabito1, Marco Stangalini2, J. Martin Laming3, Deborah Baker4,
Andy S. H. To4, David M. Long5,6, David H. Brooks7, Shahin Jafarzadeh8,9,
David B. Jess5,10, Gherardo Valori7 1INAF Istituto Nazionale di Astrofisica,
Osservatorio Astronomico di Capodimonte, 80131 Napoli, Italy
<EMAIL_ADDRESS>2ASI – Italian Space Agency, Via del Politecnico,
s.n.c 00133 – Rome, Italy  3Space Science Division, Code 7684, Naval Research
Laboratory, Washington DC 20375, USA  4University College London, Mullard
Space Science Laboratory, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
5Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, University Road, Belfast, BT7 1NN, Northern Ireland, UK
6School of Physical Sciences, Dublin City University, Glasnevin Campus,
Dublin, D09 V209, Ireland  7Department of Physics & Astronomy, George Mason
University, 4400 University Drive, Fairfax, VA 22030, USA  8Max Planck
Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen,
Germany  9Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej
17, DK-2100 Copenhagen, Denmark  10Department of Physics and Astronomy,
California State University Northridge, Northridge, CA 91330, USA

###### Abstract

We investigate the propagation of Alfvén waves in the solar chromosphere,
distinguishing between upward and downward propagating waves. We find clear
evidence for the reflection of waves in the chromosphere and differences in
propagation between cases with waves interpreted to be resonant or nonresonant
with the overlying coronal structures. This establishes the wave connection to
coronal element abundance anomalies through the action of the wave
ponderomotive force on the chromospheric plasma, which interacts with
chromospheric ions but not neutrals, thereby providing a novel mechanism of
ion-neutral separation. This is seen as a “First Ionization Potential Effect”
when this plasma is lifted into the corona, with implications elsewhere on the
Sun for the origin of the slow speed solar wind and its elemental composition.

Usage

Structure

††preprint: APS/123-QED

The first hint of varying elemental composition in the solar atmosphere came
in 1963 [1]. Taken seriously in the 1980’s following influential reviews [2,
3], the phenomenon was systematized as an enhancement in coronal abundances by
about a factor of 3 in elements such as Fe, Mg, and Si with first ionization
potential (FIP) less than about 10 eV [4, 5]. High-FIP elements, those with
FIP greater than 10 eV (e.g. O, Ne, He, and S), are relatively unaffected.
This fractionation has become known as the “FIP Effect”.

In the mid 1990s, imaging time series of solar atmospheric dynamics revealed
significantly larger-than-anticipated wave motions in the corona, e.g. [7, 6].
In the absence of an explanation for the FIP Effect in terms of “conventional”
processes such as the various forms of diffusion or gravitational settling,
this last observation suggested a role for magnetohydrodynamic (MHD) waves
interacting with plasma ions (but not with neutrals) via the ponderomotive
force [8]. The basic mechanism involved a coronal loop that acts as a resonant
cavity for MHD waves, while wave reflection/refraction in the high density
gradient in the solar chromosphere can separate the ions (low-FIP elements)
from the neutrals (high-FIP elements). This new feature in solar physics is an
analogy in MHD of the trapping of particles in optical physics by the
refraction of laser beams [9, 10].

Observations of elemental abundances now encompass a wide variety of coronal
regions and solar wind regimes [11, 12, 13, 14, 15, 16, 17]. These different
fractionations can be reproduced by the ponderomotive force model, where
simple changes to the magnetic geometry in the chromosphere-corona region
introduce modifications to the MHD wave fields, which in turn alter the
fractionation in realistic ways. For example, the reduced FIP fractionation in
the fast solar wind from polar coronal holes compared to the slow wind from
more equatorial solar regions is easily reproduced by modern models [18]. It
has also been well established that turbulence in the fast solar wind observed
close to Earth orbit [19, 20] is more Alfvénic and less balanced (i.e. more
outgoing than ingoing waves) than in the slow wind. Along with the abundance
modification, if caused by waves, these properties too should be seeded by the
same waves connected to the fractionation region.

Figure 1: Panel (a): Full-disk SDO/AIA 193 Å coronal map with selected field
lines from the PFSS extrapolation based on a SDO/HMI synoptic magnetogram,
showing three populations: small loops (black), big loops (yellow), and open
field lines (white). Panel (b): 3D overview of the region of interest linking
the chromospheric disturbances (“Blue Dots”) and locations of strong coronal
FIP-bias through a linear force free extrapolation, rotated by nearly 90∘ for
better visualization. Panel (c): Chromospheric DST/IBIS Ca II 8542.2 nm line-
core intensity image showing Blue Dots and Box A locations. Panel (d):
Chromospheric velocity amplitude of Alfvén(ic) waves. The hatched area shows
where the Ca II Stokes $I$ intensity profile goes into emission. Adapted from
[22].

Thanks to a combination of high-resolution ground-based spectropolarimetric
observations of the solar chromosphere and space observations of the solar
corona, regions of strong coronal FIP bias were linked for the first time to
the existence of chromospheric magnetic waves in a sunspot [21, 22]. So far,
the existence of MHD wave reflection/refraction in the chromosphere where FIP
fractionation happens has only been theorized, and their cross-helicity
(“balance”) only measured in situ in the solar wind. In this work we study
Alfvén wave balance and reflection in the chromosphere, and reinforce our
ideas about FIP fractionation.

The target of this study was acquired with the Interferometric BIdimensional
Spectrometer [IBIS; 23] instrument at the Dunn Solar Telescope (DST) on 2016
May 20 for more than two hours of observations. The same portion of this
active region (i.e., the large sunspot) was also observed using the EUV
Imaging Spectrometer [EIS; 24] on board the _Hinode_ satellite [25]. These
specific data were used to make a spatially resolved map of coronal
composition. In detail, the ratio of the low-FIP Si x 258.375Å to the high-FIP
S x 264.223Å diagnostic was used to compute the FIP bias map [22]. The ground-
based observations are stored in the IBIS-A archive [26] and can be accessed
at http://ibis.oa-roma.inaf.it/IBISA.

Figure 2:  Phase lag diagrams between 300 km and 1300 km in the solar
atmosphere from the Ca ii 854.2 nm Stokes $V$ (the first eleven spectral
positions) in the two regions BD (left) and Box A (right) as identified in
Figure 1c. All plots are obtained by considering only those phase values for
which the coherence is larger than 90% in each frequency bin. Modelled phase
shifts are shown from lightest to darkest red for both BD and Box A. These use
slightly different height intervals to the observations (see text), but the
qualitative agreement is clear. The shaded area indicates the change in the
propagation of the waves.

An overview of the observations described above is presented in Figure 1.
Panel (a) shows a full disk coronal image with a superimposed potential field
source surface (PFSS) extrapolation. Within the limits of the approximation
[27, 28, 29] this should give a good representation of the large scale solar
magnetic field structure. Panel (b), adapted from [22], shows a coronal FIP
bias map. The western side of the sunspot shows a FIP bias of between 1.5 and
2. Here, a linear force free extrapolation shows that this region (“Box A”) is
magnetically connected to another active region or has open field lines, the
yellow and white lines in panel (a), respectively. By contrast higher FIP bias
values of 3-4 are found on the eastern/southern sides of the spot in the
active region core loops (black field lines), connected with locations of
enhanced chromospheric wave power, shown as “Blue Dots” (BDs). These are
overplotted on the ground-based high-resolution Ca ii 854.2 nm line-core
intensity image in panel (c). Panel (d) shows the velocity amplitude of the
Alfvénic waves. The strong Alfvénic perturbations in the chromosphere [21] are
magnetically linked to coronal loops containing highly fractionated plasma
[22] as expected if waves cause the fractionation.

The ground-based observations consist of full Stokes ($I$, $Q$, $U$, and $V$
measurements) scans of the Ca ii 854.2 nm lines, acquired along with 21
spectral points with a cadence of 48 s. Taking into account the Ca ii
formation height, i.e., from line wing to line core, the 21 spectral points
sample the solar atmosphere spanning from the mid-photosphere ($\sim$300 km
formation height, on average) to the mid-chromosphere [$\sim$1300 km formation
height; 30]. These multi-height observations allow us to investigate wave
activity through the solar atmosphere [31]. Using a Fourier phase-lag
technique (see the Appendix [32]) it is possible, by employing a stringent
coherence confidence level, to retrieve important wave characteristics as they
propagate between various layers of the solar atmosphere.

In order to perform this task, we calculate the phase angles and associated
coherence levels over the whole frequency spectrum between pairs of Ca ii
854.2 nm Stokes $V$ (i.e., linked to the longitudinal magnetic field) through
consecutive spectral points. From the first 9 measured spectral points, we are
able to explore the coupling between the upper photosphere and lower
chromosphere [33, 34]. To have reliable phase measurements, we only include
those that have a coherence level greater than 90%, which ensures the accuracy
and statistical significance of our identified waves [31]. We are helped by
the unusually strong magnetic field in the sunspot, taken to be 1500 G in the
penumbra chromosphere [30, 35].

Figure 2 shows the phase-lag diagrams for these two regions (i.e., BD and Box
A on the left and right side, respectively) using the Stokes $V$ time series.
In each panel, the curves are plotted from the lightest to darkest colors,
associated with the lowest to highest solar atmospheric heights (i.e., 300 km
above the solar surface up to 1300 km in the solar chromosphere). The
geometric-height approximations, corresponding to the spectral points, were
derived using the RH code [36], assuming the FALC atmospheric model [32, 37],
to be consistent with the NLTE inversion results used here and in previous
works. While the FALC atmospheric model assumes a quiet Sun region, the
difference between the chromosperic structure in quiet Sun and sunspot regions
do not contribute to significant differences in the derived heights of each of
the spectral points. Waves travel upwards/downwards when positive/negative
phases are detected, respectively. All the magnetic waves (considering the
Stokes $V$ measurements) detected inside Box A (right panel of Figure 2)
travel upwards from the photosphere into the chromosphere. This is _not_ the
case within the BD region, where the low frequency waves show positive phases
and move upwards, while negative phases, indicating downward motion, for
frequencies $>4$ mHz are detected at upper-atmospheric heights (958 – 1393 km;
darker blue lines in the left panel of Figure 2). This indicates a source of
waves higher up in the atmosphere, or possibly in the opposite chromosphere
(see the red dots or the shaded area in the left panel of Figure 2).

Figure 3:  Wave propagation models used to interpret observations. Panel
(a):FALC chromosphere density and temperature profiles. Panels (b) and (e):
Wave amplitudes and energy fluxes for the Blue Dots (BDs) region. In the order
of decreasing wave amplitude or flux, the wave frequencies are 3.04, 5.55, and
8.3 mHz. Panels (c) and (f): Same plots (as in (b) and (e)) for waves in Box
A, where these frequencies represent the loop fundamental, as well as 1st and
2nd harmonics. Panel (d): Si/S fractionation resulting from the ponderomotive
force in each region. The discontinuities in all panels at 2250 km arise from
the steep chromosphere-corona density gradient.

In the most developed models of the FIP fractionation, [8, 18, 38, 39], ion-
neutral separation arises in the chromosphere as a result of the time
independent part of the ponderomotive force associated with the passage of
Alfvén and/or fast mode waves. This force is dictated by the wave field
profile as a result of wave propagation in the structured atmosphere, with
reflection due to density and magnetic-field gradients, as well as nonlocal
effects arising as the waves interact with coronal structures (most easily
characterized in terms of resonance). A wave is considered resonant with a
coronal loop if the wave travel-time from one loop footpoint (a location of
strong wave reflection) to the other is an integral number of wave half-
periods.

The PFSS extrapolation reveals three populations of magnetic structures
connecting to the umbra. There are small core loops on the east side, depicted
in black, connecting two footpoints within the same active region. These have
lengths, $L$, of order of 10–100 Mm and magnetic fields of 50–500 G
(comparable with that reported by [40] using the observations of Fe x).
Together with coronal densities, calculated using the observed EIS Fe xiii
[22], around $10^{9}$ cm-3, these yield resonant frequencies, $\nu=V_{A}/L$,
where $V_{A}$ is the Alfvén speed, in the range of 0.1–1 Hz. Longer loops,
shown in yellow, connect the western side of the sunspot to other active
regions and are about 1000–2000 Mm long, with magnetic fields $\sim 10$ G,
that, with coronal densities in the $10^{7}-10^{8}$ cm-3 range give loop
resonant frequencies of 1–10 mHz, i.e., the range observed in the present
work. Finally open fields, identified by white curves, are less relevant to
this study, and may actually be part of the population of longer loops that
are sufficiently long to leave the computational domain.

We construct models of the wave propagation presented in Figure 3. Waves are
excited at about 1000 km altitude in one chromosphere, and the transport
equations [32] are integrated to the other footpoint to give the behavior
there as displayed. Panel (a) shows the density and temperature profiles from
the chromospheric model FALC [37], the same model used in the NLTE
spectropolarimetric inversions reported in [30]. Panel (d) illustrates the FIP
fractionation for Si/S for both the BD and Box A. Panels (b) and (e)
respectively display the wave amplitudes and energy fluxes for the BD region.
In the order of decreasing amplitude or flux, the wave frequencies are 3.04,
5.55, and 8.3 mHz. Panels (c) and (f) show the same plots as in (b) and (e),
but for waves in Box A, where their associated frequencies represent the loop
fundamental, as well as the 1st and 2nd harmonics. At chromospheric heights of
1500 km, the directionality of the waves shown in Figure 3 panel (e) and (f)
agrees with that observed in Figure 2 for both the BD and Box A regions.
Furthermore Figure 3 panel (d) provides the observed FIP fractionation for
each case. The ponderomotive acceleration has been calculated from the Alfvén
wave amplitudes with parametric generation of sound waves taken into account,
following [18]. At lower atmospheric heights, around 500–1200 km, the
directionality is preserved better in the 3.04 mHz waves, and is reduced for
the higher frequencies, as also in the observations. At even lower geometric
heights, all waves have approximately equal upward and downward components, in
agreement with the observations. Phase shifts of the magnetic field
perturbations in the models are plotted with red symbols on Figure 2, for the
height intervals 180-210, 210-252, 252-276, 276-1385 km for Box A and 180-210,
210-252, 252-560, 560-1385 km for the BDs. It is worth noting that for a given
matching the frequency range between observations and our models, the
resulting height intervals are slightly different. This reflects the
difference in reflection properties between the model and real sunspot
chromospheres. However these heights are all well below the chromospheric
altitude where the fractionation occurs in our models, and so no serious error
is expected to result. Fiducial representations of the solar atmosphere at the
time of the observations are not possible since we do not have corresponding
observations of the opposite loop footpoints nor of the loop coronal sections.
Wave damping is neglected, but both the parametric generation of sound wave
and the turbulent cascade of counter propagating Alfvén wave should be
expected, especially in the chromospheric regions below 1000 km where the
Alfvén speed decreases (approaching the sound speed) and the magnetic field
lines become more curved, increasing the perpendicular component of the
wavevector.

As well as Alfvén wave propagation, the FIP fractionation depends on the
ionization balance for various elements. In the cases of H and Ca, these are
shown not to vary dramatically from their values in static chromospheric
models [41, 42]. For fractionation at the top of the chromosphere, where the
density gradients are steep and the background gas is ionized, no
fractionation happens at all unless the element neutral fraction is less than
about 0.5% (the ratio of scattering rates of neutrals and ions with the
background gas) [39]. Ionization beyond the singly charged state makes no
difference because the ponderomotive acceleration for low frequency shear or
torsional Alfvén waves is the same for all charged particles.

The resonant frequencies of the coronal structures in Box A (i.e., 1–10 mHz)
are in the range detectable with IBIS observations. We interpret these results
in the right hand panels of Figure 2 as the coronal structure picking out the
resonant frequency from a continuum of chromospheric sound waves. Except for
the swapping of downgoing and upgoing flux, the wave solutions in each
chromosphere are identical for resonant waves.

The BD loop resonances at 0.1–1 Hz are much higher than the frequencies of the
observed chromospheric waves, which are now nonresonant. We assume the same
wave frequencies as for Box A, calculated in the same way, with about twice
the amplitude as before. These waves are now off resonance, but reproduce the
observed Si/S fractionation of about 3.5. The velocity amplitudes observed in
Ca II in Fig. 1b are now much higher than in Box A in Fig. 1c, because wave
energy is not trapped in the coronal loop and drains out to the chromosphere.
As well as the FIP fractionations, the calculated wave reflection
characteristics also match well with observations.

Previous studies [21, 22, 43] highlighted the existence of magnetic
oscillations as a driver for the coronal fractionation. This aspect has always
been the key ingredient of the ponderomotive force introduced for separating
ions from neutrals in the chromosphere. We go further in this work and infer
that the magnetic oscillations, detected and linked with the high FIP bias,
are directional in the chromosphere and corona as predicted by models of the
fractionation. The resonant waves are more balanced in the coronal sections of
the loop than are the nonresonant waves, suggesting that as well as the
abundances, the different turbulence properties in various regimes of the
solar wind might be seeded by their solar origins. We emphasise that the wave
properties are dictated by conditions at each end of the coronal loop and the
degree to which the wave is resonant with the loop, and so the fractionation
is inherently nonlocal.

###### Acknowledgements.

MM is currently at INAF Istituto Nazionale di Astrofisica, Osservatorio
Astronomico di Roma, 00078, Monteporzio Catone, Space Science Data Center
(SSDC) - Agenzia Spaziale Italiana, via del Politecnico, s.n.c., I-00133,
Roma, Italy. This work was supported by the Italian Space Agency (ASI) with
the scientific contribution to Solar-C under contract to the co-financing
National Institute for Astrophysics (INAF) Accordo ASI-INAF 2021-12-HH.0
“Missione Solar-C EUVST – Supporto scientifico di Fase B/C/D”. MM acknowledge
financial support from the ASI-INAF agreement n. 2022-14-HH.0. JML was
supported by NASA HSR Grant NNH22OB102 and by Basic Research Funds of the
Office of Naval Research. ASHT thanks the STFC for support via funding given
in his PhD studentship. DML is grateful to the UK Science Technology and
Facilities Council (STFC) for the award of an Ernest Rutherford Fellowship
(ST/R003246/1). DBJ acknowledges support from the UK Space Agency for a
National Space Technology Programme (NSTP) Technology for Space Science award
(SSc 009). DBJ is also grateful to the UK STFC for additional funding via the
grant awards ST/T00021X/1 and ST/X000923/1. DBJ also wishes to thank The
Leverhulme Trust for grant RPG-2019-371. GV acknowledges funding by the
Bundesministerium für Wirtschaft und Technologie through Deutsches Zentrum für
Luft- und Raumfahrt e.V. (DLR), Grants No. 50 OT 1001/1201/1901 as well as 50
OT 0801/1003/1203/1703, and by the President of the Max Planck Society (MPG).
Hinode is a Japanese mission developed and launched by ISAS/JAXA,
collaborating with NAOJ as a domestic partner, and NASA and STFC (UK) as
international partners. Scientific operation of Hinode is performed by the
Hinode science team organized at $ISAS/JAXA$. This team mainly consists of
scientists from institutes in the partner countries. Support for the post-
launch operation is provided by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA,
and NSC (Norway). DB is funded under STFC consolidated grant number
ST/S000240/1. The work of DHB was performed under contract to the Naval
Research Laboratory and was funded by the NASA Hinode program. Finally, we
wish to acknowledge scientific discussions with the Waves in the Lower Solar
Atmosphere (WaLSA; https://www.WaLSA.team) team, which has been supported by
the Research Council of Norway (project no. 262622), The Royal Society [award
no. Hooke18b/SCTM; 44], and the International Space Science Institute (ISSI
Team 502).

Author contributions – Conceptualization: M.M., M.S., D.B.; Formal Analysis:
M.M., J.M.L., A.S.H.To, S.J.; Investigation, Methodology, Project
administration, Resources, Software, Visualization: M.S.; Writing-original
draft: All; Writing-review and editing: All.

## References

  * Pottasch [1963] S. R. Pottasch, Astrophys. J. 137, 945 (1963). doi:10.1086/147569
  * Meyer [1985] J.-P. Meyer, Astrophys. J. Supp. 57, 151 (1985). doi:10.1086/191000
  * Meyer [1985] J.-P. Meyer, Astrophys. J. Supp. 57, 173 (1985). doi:10.1086/191001
  * Feldman & Laming [2000] U. Feldman and J. M. Laming, Physica Scripta 61, 222 (2000). doi:10.1238/Physica.Regular.061a00222
  * Feldman & Widing [2003] U. Feldman and K. G. Widing, Space Science Rev. 107, 665 (2003). doi:10.1023/A:1026103726147
  * Nakariakov et al. [1999] V. M. Nakariakov, L. Ofman, E. E. Deluca et al., Science 285, 862 (1999). doi:10.1126/science.285.5429.862
  * Thompson et al. [1999] Thompson, B. J., Gurman, J. B., Neupert, W. M., et al., Astrophys. J. Lett., 517, L151 (1999). doi:10.1086/312030
  * Laming [2004] J. M. Laming, Astrophys. J. 614, 1063 (2004). doi:10.1086/423780
  * Ashkin [1970] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970). doi:10.1103/PhysRevLett.24.156
  * Ashkin et al. [1986] A. Ashkin, J. M. Dziedzic, J. E., Bjorkholm et al., Optics Letter, 11, 288 (1986). doi:10.1364/OL.11.000288
  * Warren [2014] H. P. Warren, Astrophys. J. Lett. 786, L2 (2014). doi:10.1088/2041-8205/786/1/L2
  * Baker et al. [2013] D. Baker, D. H. Brooks, P. Démoulin et al., Appl. Phys. , 778, 69 (2013). doi:10.1088/0004-637X/778/1/69
  * Ko et al. [2016] Y.-K. Ko, P. R. Young, K. Muglach et al., Astrophys. J. 826, 126 (2016). doi:10.3847/0004-637X/826/2/126
  * Feldman et al. [1998] U. Feldman, U. Schühle, K. G. Widing et al., Astrophys. J. 505, 999 (1998). doi:10.1086/306195
  * Baker et al. [2018] D. Baker, D. H. Brooks, L. van Driel-Gesztelyi et al., Astrophys. J. 856, 71 (2018). doi:10.3847/1538-4357/aaadb0
  * von Steiger et al. [2000] R. von Steiger, N. A. Schwadron, L. A., Fisk et al., J. Geophys. Res. 105, 27217 (2000). doi:10.1029/1999JA000358
  * Zurbuchen et al. [2012] T. H. Zurbuchen, R. von Steiger, J. Gruesbeck et al., Space Science Rev. 172, 41 (2012). doi:10.1007/s11214-012-9881-5
  * Laming [2015] J. M. Laming, Living Reviews in Solar Physics 12, 2 (2015). doi:10.1007/lrsp-2015-2
  * Bruno & Carbone [2013] R. Bruno and V. Carbone, Living Reviews in Solar Physics 10, 2 (2013). doi:10.12942/lrsp-2013-2
  * Ko et al. [2018] Y.-K. Ko, D. A. Roberts and S. T. Lepri, Astrophys. J. 864, 139 (2018). doi:10.3847/1538-4357/aad69e
  * Stangalini et al. [2021] M. Stangalini, D. Baker, G. Valori et al., Philosophical Transactions of the Royal Society of London Series A 379, 20200216 (2021). doi:10.1098/rsta.2020.0216
  * Baker et al. [2021] D. Baker, M. Stangalini, G. Valori et al., Astrophys. J. 907, 16 (2021). doi:10.3847/1538-4357/abcafd
  * Cavallini [2006] F. Cavallini, Solar Phys, 236, 415 (2006). doi:10.1007/s11207-006-0103-8
  * Culhane et al. [2007] J. L. Culhane, L. K. Harra, A. M. James et al., Solar Phys. 243, 19 (2007). doi:10.1007/s01007-007-0293-1
  * Kosugi et al. [2007] T. Kosugi, K. Matsuzaki, T. Sakao et al., Solar Phys. 243, 3 (2007). doi:10.1007/s11207-007-9014-6
  * Ermolli et al. [2022] I. Ermolli, F. Giorgi, M. Murabito et al., Astron. Astrophys. 661, A74 (2022). doi:10.1051/0004-6361/202142973
  * Riley et al. [2006] P. Riley, J. A. Linker, Z. Mikić, R. Lionello, S. A. Ledvina, and J. G. Luhmann, Astrophys. J. , 653, 1510 (2006). doi:10.1086/508565
  * Linker et al. [2017] J. A. Linker, R. M. Caplan, C. Downs, et al., Astrophys. J. , 848, 70 (2017). doi:10.3847/1538-4357/aa8a70
  * Loeschl et al. [2024] P. Loeschl, G. Valori, J. Hirzberger, et al., A& A, 681, A59 (2024). doi:10.1051/0004-6361/202346046
  * Murabito et al. [2019] M. Murabito, I. Ermolli, F. Giorgi et al., Astrophys. J. 873, 126 (2019). doi:10.3847/1538-4357/aaf727
  * Jess et al. [2023] D. B. Jess, S. Jafarzadeh, P. H. Keys et al., Living Reviews in Solar Physics 20, 1 (2023). doi:10.1007/s41116-022-00035-6
  * Suppl [2023] See Supplemental Material at … for brief dicussion.
  * de la Cruz Rodríguez et al. [2013] J. la Cruz Rodríguez, B. De Pontieu, M. Carlsson et al., Astrophys. J. Lett. 764, L11 (2013). doi:10.1088/2041-8205/764/1/L11
  * Bjørgen et al. [2018] J. P. Bjørgen, A. V. Sukhorukov, J. Leenaarts et al., Astron. Astrophys. 611, A62 (2018). doi:10.1051/0004-6361/201731926
  * Stangalini et al. [2022] Stangalini, M., Verth, G., Fedun, V., et al., Nature Communications 13, 479, (2022). doi:10.1038/s41467-022-28136-8
  * Uitenbroek [2001] H. Uitenbroek, Astrophys. J. 557, 389 (2001). doi:10.1086/321659
  * Fontenla et al. [1993] J. M. Fontenla, E. H. Avrett and R. Loeser, Astrophys. J. 406, 319 (1993). doi:10.1086/172443
  * Laming [2017] J. M. Laming, Astrophys. J. 844, 153 (2017). doi:10.3847/1538-4357/aa7cf1
  * Laming et al. [2019] J. M. Laming, A. Vourlidas, C. Korendyke et al., Appl. Phys. , 879, 124 (2019). doi:10.3847/1538-4357/ab23f1
  * Brooks et al. [2021] D. H. Brooks, H. P. Warren and E. Landi, Astrophys. J. Lett. 915, L24 (2021). doi:10.3847/2041-8213/ac0c84
  * Carlsson & Stein [2002] M. Carlsson & R. F. Stein, Astrophys. J. 572, 626, (2002). doi:10.1086/340293
  * Wedermeyer-Böhm & Carlsson [2011] S. Wedermeyer-Böhm, S. & M. Carlsson, Astron. Astrophys. 529, 9, (2011). doi:10.1051/0004-6361/201016186
  * Murabito et al. [2021] M. Murabito, M. Stangalini, D. Baker, et al., Astron. Astrophys. 656, A87 (2021). doi:10.1051/0004-6361/202141504
  * Jess et al. [2021] D. B. Jess, P. H. Keys, M. Stangalini et al., Philosophical Transactions of the Royal Society of London Series A 379, 20200169 (2021). doi:10.1098/rsta.2020.0169
  * Bendat and Peirsol [2000] J.S. Bendat and A. G. Peirsol, BOOK REVIEW: Random Data Analysis and Measurement Procedures, Measurement Science and Technology 11, 1825 (2000).
  * Penn et al. [2011] M. J. Penn, T. Schad, and E. Cox, Astrophys. J. 734, 47 (2011). doi:10.1088/0004-637X/734/1/47
  * Vaughan and Nowak [1997] B. A. Vaughan and M. A. Nowak, Astrophys. J. Lett. 474, LL43 (1997). doi:10.1086/310430
  * Rybicki and Hummer [1991] G. B. Rybicki and D. G. Hummer, Astron. Astrophys. 245, 171 (1991).
  * Rybicki and Hummer [1992] G. B. Rybicki and D. G. Hummer, Astron. Astrophys. 262, 209 (1992).
  * Reardon and Cavallini [2003] K. Reardon and F. Cavallini, Mem. Soc. Astron. Italiana 74, 815 (2003).
  * Fontenla et al. [2009] J. M. Fontenla, W. Curdt, M. Haberreiter et al., Astrophys. J. 707, 482 (2009). doi:10.1088/0004-637X/707/1/482
  * Laming [2021] J. M. Laming, Astrophys. J. 909, 17 (2021). doi:10.3847/1538-4357/abd9c3


# Subdivisions with congruence constraints in digraphs of large chromatic
number

Raphael Steiner

###### Abstract.

We prove that for every digraph $F$ and every assignment of pairs of integers
$(r_{e},q_{e})_{e\in A(F)}$ to its arcs there exists an integer $N$ such that
every digraph $D$ with dichromatic number greater than $N$ contains a
subdivision of $F$ in which $e$ is subdivided into a directed path of length
congruent to $r_{e}$ modulo $q_{e}$, for every $e\in A(F)$.

This generalizes to the directed setting the analogous result by Thomassen
[15] for undirected graphs, and at the same time yields a novel short proof of
his result.

Department of Computer Science, Institute of Theoretical Computer Science, ETH
Zürich, Switzerland<EMAIL_ADDRESS>The author was
supported by an ETH Zurich Postdoctoral Fellowship.

##  1\. Introduction

Given a graph $F$, a _subdivision_ of $F$ is a graph $H$ consisting of $v(F)$
pairwise distinct so-called _branch-vertices_ $(b_{f})_{f\in V(F)}$ and a
collection of pairwise internally disjoint so-called _subdivision paths_
$(P_{e})_{e\in E(F)}$, each of positive length, such that $P_{e}$ has
endpoints $b_{u}$ and $b_{v}$ for every edge $e=uv$ in $F$. The exact same
definition applies to directed graphs, with the sole difference being that for
every arc $e=(u,v)$ in a digraph $F$ we require the corresponding subdivision-
path $P_{e}$ to be directed from $b_{u}$ to $b_{v}$.

The study of subdivision containment in undirected an directed graphs has a
long history in graph theory. Among the seminal results in the area is a
famous theorem of Mader [13], stating that for every $k\in\mathbb{N}$ there is
a number $d(k)$ such that every graph of average degree at least $d(k)$
contains a subdivision of $K_{k}$ (and hence, of any other graph of order at
most $k$). While Mader’s bound on $d(k)$ was exponential, the exact order of
magnitude of $d(k)$ was later determined to be $d(k)=\Theta(k^{2})$ by
Bollobás and Thomason [4] and independently by Komlós and Szemerédi [12].
Since every graph $G$ contains a subgraph of minimum degree at least
$\chi(G)-1$, the above also implies that chromatic number at least
$\Theta(k^{2})$ forces the containment of a $K_{k}$-subdivision.

A strengthening of Hadwiger’s conjecture, the so-called _Hajós conjecture_ ,
claimed that in fact every graph of chromatic number at least $k$ contains a
$K_{k}$-subdivision, but this is now known to be false. Explicit
counterexamples for $k\geq 7$ were constructed by Catlin [5] and later, Erdős
and Fajtlowicz [8] showed that there exist graphs without a
$K_{k}$-subdivision and chromatic number at least $c\frac{k^{2}}{\log k}$ for
some absolute constant $c>0$.

The above results guarantee that for every fixed graph $F$, any graph $G$ of
sufficiently large chromatic number will contain an $F$-subdivision. However,
they give no control over the path lengths in this subdivision. In particular,
the found subdivision may be a bipartite graph. In contrast to this, we know
that every graph of chromatic number at least $3$ contains an odd cycle, and
in fact it is even known that graphs of sufficiently large chromatic number
contain cycles of length $r$ modulo $q$, for every given pair of integers $r$
and $q$, see for instance [3, 15, 6]. This naturally raises the question
whether additional congruence constraints may be enforced on the path lengths
in an $F$-subdivision in graphs of large chromatic number. Related to this,
Toft conjectured in 1975 that every graph of chromatic number at least $4$
contains a _totally odd subdivision_ of $K_{4}$, that is, a subdivision with
all subdivision paths of odd length. This conjecture was verified in the late
90s independently by Zang [17] and Thomassen [16]. A qualitatively more
powerful result was established by Thomassen already in 1983, as follows.

######  Theorem 1 (Thomassen, Theorem 12 in [15]).

Let $F$ be a graph, and for every $e\in E(F)$ let $r_{e},q_{e}\in\mathbb{Z}$
be integers with $q_{e}\geq 2$.

Then there exists a number $N\in\mathbb{N}$ such that every graph $G$ with
$\chi(G)\geq N$ contains a subdivision of $F$ such that for every $e\in E(F)$,
the corresponding subdivision-path is of length congruent to $r_{e}$ modulo
$q_{e}$.

This result largely resolves the question posed above in the positive. We
refer to [2, 11] for some more recent results on subdivisions with congruence-
constraints on the path lengths.

In this note we prove a generalization of Theorem 1 to directed graphs.
Accordingly, we need to specify a notion of chromatic number for directed
graphs. Here, we will stick with the recently popular notion of the
_dichromatic number_ , which was introduced by Erdős [7] and Neumann-Lara [14]
in 1980. Given a digraph $D$, the _dichromatic number_ of $D$ (denote by
$\vec{\chi}(D)$) is the smallest integer $k\geq 1$ such that $V(D)$ can be
partitioned into $k$ sets $V_{1},\ldots,V_{k}$, each of them inducing an
acyclic subdigraph of $D$.

At first sight, it may not be obvious at all why sufficiently large
dichromatic number forces the containment of subdivisions in directed graphs.
However, using an elegant argument, this fact was recently established by
Aboulker, Cohen, Havet, Lochet, Moura and Thomassé [1], see also [9, 10] for
related results.

######  Theorem 2 (Aboulker et al. [1]).

For every digraph $F$ there exists a number $N\in\mathbb{N}$ such that every
digraph $D$ with $\vec{\chi}(D)\geq N$ contains a subdivision of $F$.

Our main result stated below is a qualitative strengthening of Theorem 2,
additionally ensuring the path lengths in the subdivision of $F$ to satisfy
given congruences.

######  Theorem 3.

Let $F$ be a digraph, and for every $e\in A(F)$ let $r_{e},q_{e}\in\mathbb{Z}$
be integers with $q_{e}\geq 2$.

Then there exists a number $N\in\mathbb{N}$ such that every digraph $D$ with
$\vec{\chi}(D)\geq N$ contains a subdivision of $F$ such that for every $e\in
A(F)$, the corresponding subdivision-path is of length congruent to $r_{e}$
modulo $q_{e}$.

Theorem 3 is the natural analogue of Thomassen’s Theorem 1 for directed
graphs. In fact, it also easily implies Thomassen’s theorem, which can be seen
as follows:

Let $F$ be any given graph equipped with integer pairs $(r_{e},q_{e})_{e\in
E(F)}$. Fix an orientation $\vec{F}$ of $F$. Then by Theorem 3 there exists
$N\in\mathbb{N}$ such that any digraph $D$ with $\vec{\chi}(D)\geq N$ contains
a subdivision of $\vec{F}$ in which every oriented edge $e$ in $\vec{F}$ is
replaced by a directed path of length congruent to $r_{e}$ modulo $q_{e}$.
Now, let $G$ be any given undirected graph with $\chi(G)\geq N$, and let
$\accentset{\leftrightarrow}{G}$ denote the directed graph on the same vertex-
set as $G$ with arc-set $\\{(u,v),(v,u)|uv\in E(G)\\}$. Pause to note that
$\vec{\chi}(\accentset{\leftrightarrow}{G})=\chi(G)\geq N$. But then by the
above $\accentset{\leftrightarrow}{G}$ must contain a subdivision of $\vec{F}$
with path-lengths satisfying the congruence-constraints given by the sequence
$(r_{e},q_{e})_{e\in A(\vec{F})}$. This translates (by ignoring the directions
of edges) one-to-one to a subdivision of $F$ in $G$ in which every edge $e$ is
replaced by a path of length $r_{e}$ modulo $q_{e}$.

We may conclude that Theorem 1 is a special case of Theorem 3. In addition,
our short proof of Theorem 3 is conceptually entirely different from
Thomassen’s proof of Theorem 1 in [15], which may be interesting in its own
right.

As a last point, let us mention that by considering the special case when $F$
is a directed cycle of length two, the proof of Theorem 3 also implies that
there exists a (reasonably small) absolute constant $C>0$ such that every
digraph of dichromatic number greater than $Cq$ contains a directed cycle of
length congruent to $r$ modulo $q$, for every integer $r$. This resembles the
main result of Chen, Ma and Zang in [6], who proved this statement with the
optimal constant $C=1$.

### Terminology and notation.

All graphs and directed graphs considered in this paper are simple, i.e., have
no loops or parallel edges. For a graph $G$ we denote by $V(G),E(G)$ the set
of vertices and edges. For a digraph $D$ we denote by $V(D)$ the vertex-set
and by $A(D)\subseteq\\{(u,v)\in V(D)\times V(D)|u\neq v\\}$ the arc-set. We
also use the notation $a(F):=|A(F)|$ for the number of arcs. For $X\subseteq
V(D)$ we denote by $D[X]$ the subdigraph of $D$ consisting of the vertex-set
$X$ and all arcs of $D$ going between vertices of $X$. Given a directed path
$P$ or a directed cycle $C$ in $D$, we denote by $\ell(P)$ resp. $\ell(C)$ its
length. Paths are allowed to consist of a single vertex (i.e., have length
zero). We say that $D$ is strongly connected if for every $(x,y)\in V(D)^{2}$
there exists a directed path from $x$ to $y$ in $D$. A _strongly connected
component_ of $D$ is a maximal subset $X$ of vertices such that $D[X]$ is
strongly connected. For vertices $x,y$ in a strongly connected digraph $D$ we
denote by $\text{dist}_{D}(x,y)$ the length of a shortest directed path from
$x$ to $y$.

##  2\. Proof of Theorem 3

The proof of Theorem 2 by Aboulker et al. was based on the insight that BFS
layerings in directed graphs can be used to find vertex-disjoint directed
paths for a subdivision. Also our proof of Theorem 3 will rely crucially on
the BFS layering idea. In particular, we need Lemma 4 and Lemma 5 stated below
which both appear (in equivalent form) in [1].

######  Lemma 4 (cf. [1], Lemma 29).

The dichromatic number of a digraph $D$ equals the maximum of the dichromatic
numbers of the subdigraphs induced by its strongly connected components.

######  Lemma 5 (cf. [1], Lemma 30).

Let $D$ be a strongly connected digraph and let $v\in V(D)$. For every integer
$i\geq 0$, let $L_{i}^{+}=L_{i}^{+}(v,D):=\\{x\in
V(D)|\text{dist}_{D}(v,x)=i\\}$ and $L_{i}^{-}=L_{i}^{-}(v,D):=\\{x\in
V(D)|\text{dist}_{D}(x,v)=i\\}$. Then

$\vec{\chi}(D)\leq 2\max\\{\vec{\chi}(D[L_{i}^{+}])|i\geq 0\\}$

and

$\vec{\chi}(D)\leq 2\max\\{\vec{\chi}(D[L_{i}^{-}])|i\geq 0\\}.$

We note the following simple corollary of the above.

######  Corollary 6.

Every digraph $D$ contains a set of vertices $X\subseteq V(D)$ and a vertex
$x_{0}\in X$ such that $D[X]$ is strongly connected,
$\vec{\chi}(D[X])\geq\frac{1}{2}\vec{\chi}(D)$ and for every $x\in X$ there is
a directed path $P$ in $D$ starting at $x$, ending at $x_{0}$ and such that
$V(P)\cap X=\\{x,x_{0}\\}$.

###### Proof.

By Lemma 4 we may assume without loss of generality that $D$ is strongly
connected. Pick a vertex $v$ arbitrarily and apply Lemma 5 to it. In
particular, there exists an integer $i\geq 0$ such that
$\vec{\chi}(D[L_{i}^{-}(v,D)])\geq\frac{1}{2}\vec{\chi}(D)$. Let $X$ be the
set of vertices of a strongly connected component of $D[L_{i}^{-}(v,D)]$ with
maximum dichromatic number, i.e.,
$\vec{\chi}(D[X])=\vec{\chi}(D[L_{i}^{-}(v,D)])\geq\frac{1}{2}\vec{\chi}(D)$
by Lemma 4. Let $P_{v,X}$ be a shortest directed path in $D$ starting at $v$
and ending at $X$. Then $P_{v,X}$ intersects $X$ only in its endpoint, call it
$x_{0}$. Now, for any given $x\in X$, consider a shortest directed path
$P_{x,v}$ from $x$ to $v$ in $D$ (i.e., of length exactly $i$). Since
$X\subseteq L_{i}^{-}(v,D)$, we have $V(P_{x,v})\cap X=\\{x\\}$. Hence the
union of the directed paths $P_{x,v}$ and $P_{v,X}$ forms a directed walk in
$D$ starting at $x$, ending at $x_{0}$ and such that no internal vertices of
it are contained in $X$. Hence, possibly after short-cutting this walk we
obtain a directed $x$ to $x_{0}$-path $P$ in $D$ which intersects $X$ only at
its endpoints. This concludes the proof. ∎

For our next step towards Theorem 3, namely Lemma 8 below, we need the
following known statement about long cycles in digraphs of given dichromatic
number, compare Corollary 38 in [1] or Theorem 7 in [10].

######  Lemma 7.

Let $k\geq 2$ be an integer and let $D$ be a digraph. If $\vec{\chi}(D)\geq k$
then $D$ contains a directed cycle of length at least $k$.

######  Lemma 8.

Let $D$ be a strongly connected digraph and let $x_{0}\in V(D)$,
$q\in\mathbb{N}$, $q\geq 2$. Suppose that $\vec{\chi}(D)\geq 2(q-1)$. Then
there exists a set $Y\subseteq V(D)$ such that
$\vec{\chi}(D[Y])\geq\frac{1}{4}\vec{\chi}(D)-\frac{q-1}{2}$ and an interval
$I\subseteq\mathbb{N}$ of $q$ consecutive integers such that the following
holds:

For every $y\in Y$ and every $\ell\in I$ there exists a directed path $Q$ of
length $\ell$ in $D$ which starts at $x_{0}$, ends at $y$, and satisfies
$V(Q)\cap Y=\\{y\\}$.

###### Proof.

By Lemma 5 there exists $i\geq 0$ such that
$\vec{\chi}(L_{i}^{+}(D,x_{0}))\geq\left\lceil\frac{1}{2}\vec{\chi}(D)\right\rceil$.
Let $S$ be a strongly connected component of $D[L_{i}^{+}(D,x_{0})]$ with
$\vec{\chi}(D[S])=\vec{\chi}(L_{i}^{+}(D,x_{0}))\geq\left\lceil\frac{1}{2}\vec{\chi}(D)\right\rceil$.

Further, let $Z\subseteq S$ be chosen inclusion-wise minimal such that
$\vec{\chi}(D[Z])\geq\left\lceil\frac{1}{2}\vec{\chi}(D)\right\rceil-(q-1)$.
Then the minimality of $Z$ and Lemma 4 imply that
$\vec{\chi}(D[Z])=\left\lceil\frac{1}{2}\vec{\chi}(D)\right\rceil-(q-1)$ and
that $D[Z]$ is strongly connected. It follows that $\vec{\chi}(D[S\setminus
Z])\geq\vec{\chi}(D[S])-\vec{\chi}(D[Z])\geq q-1$.

We claim that there exists a directed path $P$ in $D[S]$ of length exactly
$q-1$ such that $P$ intersects $Z$ only at its endpoint. This holds trivially
if $q=1$. If $q=2$ we have $\vec{\chi}(D[S\setminus Z])\geq 1$ and thus
$S\setminus Z\neq\emptyset$. Hence, the strong connectivity of $D[S]$ implies
the existence of an arc from $S\setminus Z$ to $Z$ in $D$. Next, suppose
$q\geq 3$. Then we may apply Lemma 7 to $D[S\setminus Y]$ with $k=q-1$ and
find a directed cycle $C$ in $D[S\setminus Y]$ of length at least $q-1$. Let
$R$ be a shortest directed path in $D[S]$ starting in $V(C)$ and ending in
$Z$. Let $r_{1}\in V(C)$ and $r_{2}\in Z$ denote the start- and endpoint of
$R$, such that $V(R)\cap V(C)=\\{r_{1}\\}$, $V(R)\cap Z=\\{r_{2}\\}$. We may
now consider the directed path in $D[S]$ obtained by concatenating the segment
of $C$ of length $q-2$ ending at $r_{1}$ with $R$. This path has length at
least $q-1$ and intersects $Z$ only at its endpoint $r_{2}$. Hence we may find
a subpath $P$ of $R$ of length exactly $q-1$ ending at $r_{2}$, as desired.

Moving on, consider the vertex $r_{2}$ in the strongly connected digraph
$D[Z]$ and apply Lemma 5 to it. We then find an integer $j\geq 0$ such that
$\vec{\chi}(D[L_{j}^{+}(r_{2},D[Z])])\geq\frac{1}{2}\vec{\chi}(D[Z])\geq\frac{1}{4}\vec{\chi}(D)-\frac{q-1}{2}$.
We now define $Y:=L_{j}^{+}(r_{2},D[Z])$ and $I:=[i+j,i+j+(q-1)]$. To conclude
the proof, let a vertex $y\in Y$ and a number $\ell\in I$ be given
arbitrarily. Write $l=i+j+\alpha$ for some $0\leq\alpha\leq q-1$, and let
$p(\alpha)\in V(P)$ denote the vertex at distance exactly $\alpha$ from
$r_{2}$ along $P$. Let $P_{1}$ be a shortest path from $x_{0}$ to $p(\alpha)$
in $D$. Since $p(\alpha)\in S\subseteq L_{i}^{+}(D,x_{0})$, we have
$\ell(P_{1})=i$. Further, let $P_{2}$ be a shortest path from $r_{2}$ to $y$
in $D[Z]$. Then $\ell(P_{2})=j$ since $y\in L_{j}^{+}(r_{2},D[Z])$. Hence the
path $Q$ formed as the union of $P_{1}$, the segment of $P$ from $p(\alpha)$
to $r_{2}$, and $P_{2}$ is a directed path of length $i+\alpha+j=\ell$ in $D$
from $x_{0}$ to $y$. Since $P_{1},P_{2}$ are shortest paths we further we have
$V(P_{1})\cap S=\\{p(\alpha)\\},V(P)\cap Z\subseteq V(R)\cap
Z=\\{r_{2}\\},V(P_{2})\cap Y=\\{y\\}$. This implies that $V(Q)\cap Y=\\{y\\}$,
as desired. This concludes the proof. ∎

We now give the proof of Theorem 3. The idea is to combine Corollary 6 and
Lemma 8 to enable induction on the number of arcs in $F$.

###### Proof of Theorem 3.

We prove the statement by induction of the number of arcs $a(F)$ in $F$. If
$a(F)=0$, then the claim holds true trivially (every digraph of dichromatic
number $N:=v(F)$ has at least $v(F)$ vertices and thus contains a copy of
$F$).

Moving on, suppose that $a(F)\geq 1$ and that we have established Theorem 3
for all digraphs containing strictly fewer arcs. Let $(r_{e},q_{e})_{e\in
A(F)}$ be a given assignment of integer pairs to the arcs of $F$. We pick an
arc $f\in A(F)$ arbitrarily. Let $N^{\prime}\in\mathbb{N}$ be a number such
that every digraph of dichromatic number at least $N^{\prime}$ contains a
subdivision of $F-f$ in which for every arc $e\in A(F)\setminus\\{f\\}$ the
corresponding subdivision path is of length congruent to $r_{e}$ modulo
$q_{e}$.

Define $N:=8N^{\prime}+4(q_{f}-1)$ and let $D$ be any given digraph with
$\vec{\chi}(D)\geq N$. Let a subset $X$ of $V(D)$ and a vertex $x_{0}\in X$ be
as guaranteed by Corollary 6, applied to $D$. Then
$\vec{\chi}(D[X])\geq\frac{1}{2}\vec{\chi}(D)$, $D[X]$ is strongly connected
and every vertex $x\in X$ can be connected to $x_{0}$ via a directed path
intersecting $X$ only at its endpoints. Since $D[X]$ is strongly connected, we
may apply Lemma 8. We thus find $Y\subseteq X$ with
$\vec{\chi}(D[Y])\geq\frac{1}{4}\vec{\chi}(D[X])-\frac{q_{f}-1}{2}\geq\frac{1}{8}\vec{\chi}(D)-\frac{q_{f}-1}{2}\geq
N^{\prime}$ and an interval $I$ of $q_{f}$ consecutive integers such that the
following holds: For every $y\in Y$ and every $\ell\in I$ there is a directed
$x_{0}$ to $y$-path of length $\ell$ in $D[X]$ intersecting $Y$ only in $y$.
By inductive assumption, there exists a subdivision of $F-f$ in $D[Y]$ such
that every $e\in A(F)\setminus\\{f\\}$ is replaced by a path of length
congruent to $r_{e}$ modulo $q_{e}$. Let $y_{1},y_{2}\in Y$ be the two branch-
vertices in this subdivision corresponding to the start- respectively endpoint
of the arc $f$ in $F$. We will now show that there exists a directed path $P$
from $y_{1}$ to $y_{2}$ in $D$ intersecting $Y$ only at its endpoints, with
$\ell(P)\equiv_{q_{f}}r_{f}$. Adding this path to the subdivision of $F-f$
then yields a subdivision of $F$ in $D$ with the desired congruence
properties.

To construct $P$, we first connect $y_{1}$ to $x_{0}$ via a directed path
$P_{0}$ with $V(P_{0})\cap X=\\{y_{1},x_{0}\\}$ (in particular, $P_{0}$ is
internally disjoint from $Y\subseteq X$). Let $\ell\in I$ be a number chosen
such that $\ell\equiv_{q_{f}}r_{f}-\ell(P_{0})$ (this is possible since $I$ is
an interval with $q_{f}$ elements). Then by our choice of $Y$ there exists a
path $Q$ in $D[X]$ from $x_{0}$ to $y_{2}$ such that $\ell(Q)=\ell$ and $Q$ is
internally disjoint from $Y$. Overall, $P_{0}\cup Q$ is a directed
$y_{1}$-$y_{2}$-path in $D$ which is internally disjoint from $Y$ and
satisfies $\ell(P_{0}\cup
Q)=\ell(P_{0})+\ell\equiv_{q_{f}}\ell(P_{0})+(r_{f}-\ell(P_{0}))=r_{f}$. Thus,
by adding this path to the subdivision of $F-f$ in $D[Y]$ we find a
subdivision of $F$ in $D$ in which every arc $e$ is replaced by a path of
length congruent to $r_{e}$ modulo $q_{e}$. This proves the inductive claim
and concludes the proof of the theorem. ∎

## References

  * [1] P. Aboulker, N. Cohen, F. Havet, W. Lochet, P. F. S. Moura and S. Thomassé. Subdivisions in digraphs of large out-degree or large dichromatic number. _Electronic Journal of Combinatorics_ , 26(3): P3.19 (2019).
  * [2] N. Alon and M. Krivelevich. Divisible subdivisions. _Journal of Graph Theory_ , 98: 623–629 (2021).
  * [3] B. Bollobás. Cycles modulo $k$. _Bulletin of the London Mathematical Society_ , 9: 97–98 (1977).
  * [4] B. Bollobás and A. Thomason. Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs. _European Journal of Combinatorics_ , 19: 883–887 (1998).
  * [5] Hajós graph coloring conjecture: variations and counterexamples. _Journal of Combinatorial Theory, Series B_ , 26(2): 268–274 (1979).
  * [6] Z. Chen, J. Ma and W. Zang. Coloring digraphs with forbidden cycles. _Journal of Combinatorial Theory, Series B_ , 115: 210–223 (2015).
  * [7] P. Erdős. Problems and results in number theory and graph theory. Proc. _Ninth Manitoba Conf. on Numerical Math. and Computing_ , 3–21 (1979).
  * [8] P. Erdős and S. Fajtlowicz. On the conjecture of Hajós. _Combinatorica_ , 1(2): 141–143 (1981).
  * [9] A. Girão, K. Popielarz and R. Snyder. Subdivisions of digraphs in tournaments. _Journal of Combinatorial Theory, Series B_ , 146: 266–285 (2021).
  * [10] L. Gishboliner, R. Steiner and T. Szábo. Dichromatic number and forced subdivisions. _Journal of Combinatorial Theory, Series B_ , 153: 1–30 (2022).
  * [11] K. Kawarabayashi. Totally odd subdivisions and parity subdivisions: structures and coloring. _Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms_ , 1013–1029 (2013).
  * [12] J. Komlós and E. Szemerédi. Topological cliques in graphs II. _Combinatorics, Probability and Computing_ , 5(1): 79–90 (1996).
  * [13] W. Mader. Homomorphieeigenschaften und mittlere Kantendichte von Graphen, _Mathematische Annalen_ , 174: 265–268 (1967) (in German).
  * [14] V. Neumann-Lara. The dichromatic number of a digraph. _Journal of Combinatorial Theory, Series B_ , 33(3): 265–270 (1982).
  * [15] C. Thomassen. Graph decomposition with applications to subdivisions and path systems modulo $k$. _Journal of Graph Theory_ , 7(2): 261–271 (1983).
  * [16] C. Thomassen. Totally odd subdivisions in $4$-chromatic graphs. _Combinatorica_ , 21: 417–443 (2001).
  * [17] W. Zang. Proof of Toft’s conjecture: Every graph containing no fully odd $K_{4}$ is $3$-colorable. _Journal of Combinatorial Optimization_ , 2: 117–188 (1998).


# The $A$-Möbius function of a finite group

Francesca Dalla Volta  Francesca Dalla Volta, Dipartimento di Matematica e
Applicazioni , University of Milano-Bicocca, Via Cozzi 55, 20126 Milano, Italy
<EMAIL_ADDRESS>and  Andrea Lucchini  Andrea Lucchini,
Università degli Studi di Padova
Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova,
Italy<EMAIL_ADDRESS>
###### Abstract.

The Möbius function of the subgroup lettice of a finite group $G$ has been
introduced by Hall and applied to investigate several different questions. We
propose the following generalization. Let $A$ be a subgroup of the
automorphism group $\operatorname{Aut}(G)$ of a finite group $G$ and denote by
$\mathcal{C}_{A}(G)$ the set of $A$-conjugacy classes of subgroups of $G.$ For
$H\leq G$ let $[H]_{A}~{}=~{}\\{~{}H^{a}~{}\mid~{}a\in~{}A\\}$ be the element
of $\mathcal{C}_{A}(G)$ containing $H.$ We may define an ordering in
$\mathcal{C}_{A}(G)$ in the following way: $[H]_{A}\leq[K]_{A}$ if $H^{a}\leq
K$ for some $a\in A$. We consider the Möbius function $\mu_{A}$ of the
corresponding poset and analyse its properties and possible applications.

##  1\. Introduction

The Möbius function of a finite partially ordered set (poset) $P$ is the map
$\mu_{P}:~{}P\times P\to~{}\mathbb{Z}$ satisfying $\mu_{P}(x,y)=0$ unless
$x\leq y,$ in which case it is defined inductively by the equations
$\mu_{P}(x,x)=1$ and $\sum_{x\leq z\leq y}\mu_{P}(x,z)=0$ for $x<y.$

In a celebrated paper [7], P. Hall used for the first time the Möbius function
$\mu$ of the subgroup lattice of a finite group $G$ to investigate some
properties of $G$, in particular to compute the number of generating
$t$-tuples of $G$. A detailed investigation of the properties of the function
$\mu$ associated to a finite group $G$ is given by T. Hawkes, I. M. Isaacs and
M. Özaydin in [8]. In that paper, the authors also consider the Möbius
function $\lambda$ of the poset of conjugacy classes of subgroups of $G,$
where $[H]\leq[K]$ if $H\leq K^{g}$ for some $g\in G$ (see [8, Section 7]). In
particular, they propose the interesting and intriguing question of comparing
the values of $\mu$ and $\lambda$.

In this paper we aim to generalize the definitions and main properties of the
functions $\mu$ and $\lambda$ to a more general contest. Let $G$ and $A$ be a
finite group and a subgroup of the automorphism group $\operatorname{Aut}(G)$
of $G$, respectively. Denote by $\mathcal{C}_{A}(G)$ the set of $A$-conjugacy
classes of subgroups of $G.$ For $H\leq G$ let
$[H]_{A}~{}=~{}\\{~{}H^{a}~{}\mid~{}a\in~{}A\\}$ be the element of
$\mathcal{C}_{A}(G)$ containing $H.$ We may define an ordering in
$\mathcal{C}_{A}(G)$ in the following way: $[H]_{A}\leq[K]_{A}$ if $H^{a}\leq
K$ for some $a\in A$; we consider the Möbius function $\mu_{A}$ of the
corresponding poset. We will write $\mu_{A}(H,K)$ in place of
$\mu_{A}([H]_{A},[K]_{A}).$ When $A=\operatorname{Inn}(G),$ we write
$\mathcal{C}(G)$ and $[H]$, in place of
$\mathcal{C}_{\operatorname{Inn}(G)}(G)$ and $[H]_{\operatorname{Inn}(G)}$.
When $A=1$, $\mu_{A}=\mu$ is the Möbius function in the subgroup lattice of
$G,$ introduced by P. Hall. In the case when $A=\operatorname{Inn}(G)$ is the
group of the inner automorphism, then $\mu_{\operatorname{Inn}(G)}=\lambda$.
Note that for any subgroup $A$ of $\operatorname{Aut}(G)$, we get
$[G]_{A}=\\{G\\}$.

In Section 2, we prove some general properties of $\mu_{A}$. In particular we
prove the following result:

######  Proposition 1.

Let $G$ be a finite solvable group. If $G^{\prime}\leq K\leq G$ and $A$ is the
subgroup of $\operatorname{Inn}(G)$ obtained by considering the conjugation
with the elements of $K,$ then $\lambda(H,G)=\mu_{A}(H,G)$ for any $H\leq G.$

To illustrate the meaning of the previous proposition, consider the following
example. Let $G=A_{4}$ be the alternating group of degree 4 and $A$ the
subgroup of $\operatorname{Inn}(G)$ induced by conjugation with the elements
of $G^{\prime}\cong C_{2}\times C_{2}.$ The posets $\mathcal{C}(G)$ and
$\mathcal{C}_{A}(G)$ are different. For example there are three subgroups of
$G$ of order 2, which are conjugated in $G$, but not $A$-conjugated. However
$\lambda(H,G)=\mu_{A}(H,G)$ for any $H\leq G.$

In Section 3, we generalize some result given by Hall in [7], about the
cardinality $\phi(G,t)$ of the set $\Phi(G,t)$ of $t$-tuples
$(g_{1},\dots,g_{t})$ of group elements $g_{i}$ such that $G=\langle
g_{1},\dots,g_{t}\rangle$. As observed by P. Hall, using the Möbius inversion
formula, it can be proved that

(1.1) $\phi(G,t)=\sum_{H\leq G}\mu(H,G)|H|^{t}.$

We generalize this formula, showing that $\phi(G,t)$ can be computed with a
formula involving $\mu_{A}$ for any possible choice of $A.$

######  Theorem 2.

For any finite group $G$ and any subgroup $A$ of $\operatorname{Aut}(G),$

$\phi(G,t)=\sum_{[H]_{A}\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)|\cup_{a\in
A}(H^{a})^{t}|.$

If $G$ is not cyclic, then $\phi(G,1)=0$, so we obtain the following equality,
involving the values of $\mu_{A}.$

######  Corollary 3.

If $G$ is not cyclic, then

$0=\sum_{[H]_{A}\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)|\cup_{a\in A}H^{a}|.$

Further generalizations are given in Section 4, where we consider the function
$\phi^{*}(G,t)$, which is an analogue of $\phi(G,t)$: actually,
$\phi^{*}(G,t)$ denotes the cardinality of the set of of $t$-tuples
$(H_{1},\dots,H_{t})$ of subgroups of $G$ such that $G=\langle
H_{1},\dots,H_{t}\rangle$. As a corollary of our formula for computing
$\phi^{*}(G,t)$, we obtain we following unexpected result.

######  Proposition 4.

Let $\sigma(X)$ denote the number of subgroups of a finite group $X.$ For any
finite group $G,$ the following equality holds:

$1=\sum_{H\leq G}\mu(H,G)\sigma(H).$

Finally, in Section 5, we consider one question originated from a result given
by Hawkes, Isaacs and Özaydin in [8]: they proved that the equality

$\mu(1,G)=|G^{\prime}|\textperiodcentered\lambda(1,G)$

holds for any finite solvable group $G$; later Pahlings [10] generalized the
result proving that

(1.2) $\mu(H,G)=|N_{G^{\prime}}(H):G^{\prime}\cap H|\cdot\lambda(H,G)$

holds for any $H\leq G$ whenever $G$ is finite and solvable. Following [6], we
say that $G$ satisfies the $(\mu,\lambda)$-property if (1.2) holds for any
$H\leq G.$ Several classes of non-solvable groups satisfy the
$(\mu,\lambda)$-property, for example all the minimal non-solvable groups (see
[6]). However it is known that the $(\mu,\lambda)$-property does not hold for
every finite group. For instance, it does not hold for the following finite
almost simple groups: $A_{9}$, $S_{9},$ $A_{10},$ $S_{10},$ $A_{11},$
$S_{11},$ $A_{12},$ $S_{12}$, $A_{13}$, $S_{13},$ $J_{2}$, $PSU(3,3)$,
$PSU(4,3)$, $PSU(5,2),$ $M_{12},$ $M_{23},$ $M_{24},$ $PSL(3,11),$ $HS$,
$\operatorname{Aut}(HS),$ $He$ $\operatorname{Aut}(H)$, $McL,$ $PSL(5,2),$
$G_{2}(4)$, $Co_{3}$, $P\Omega^{-}(8,2),$ $P\Omega^{+}(8,2).$ It is somehow
intriguing to notice that although the $(\mu,\lambda)$-property fails for the
sporadic groups $M_{12},$ $J_{2},$ $McL,$ it holds for their automorphism
groups.

We prove the following generalization of Pahlings’s result.

######  Theorem 5.

Let $N$ be a solvable normal subgroup of a finite group $G$. If $G/N$
satisfies the $(\mu,\lambda)$-property, then $G$ also satisfies the
$(\mu,\lambda)$-property.

An almost immediate consequence of the previous theorem is the following.

######  Corollary 6.

$PSU(3,3)$ is the smallest group which does not satisfy the $(\mu,\lambda)$
property.

In the last part of Section 5, we use Theorem 2 to deduce some consequences of
the $(\mu,\lambda)$-property. In particular we prove the following theorem.

######  Theorem 7.

Suppose that a finite group $G$ satisfies the $(\mu,\lambda)$-property. Then,
for every positive integer $t,$ the following equality is satisfied:

$\sum_{[H]\in\mathcal{C}(G)}\lambda(H,G)\left(\frac{|H|^{t-1}|G||G^{\prime}H|}{|G^{\prime}N_{G}(H)|}-|\cup_{a\in
A}(H^{a})^{t}|\right)=0.$

Some open questions are proposed along the paper.

##  2\. Applying some general properties of the Möbius function

Given a poset $P$, a closure on $P$ is a function $\bar{}:P\rightarrow P$
satisfying the following three conditions:

  * a)

$x\leq\bar{x}$ for all $x\in P$;

  * b)

if $x,y\in P$ with $x\leq y$, then $\bar{x}\leq\bar{y}$;

  * c)

$\bar{\bar{x}}=\bar{x}$ for all $x\in P$.

If $\,\bar{}\,$ is a closure map on $P,$ then $\overline{P}=\left\\{x\in
P|\,\bar{x}=x\right\\}$ is a poset with order induced by the order on $P$. We
have:

######  Theorem 8 (The closure theorem of Crapo [3]).

Let $P$ be a finite poset and let $\bar{}:P\to P$ be a closure map. Fix
$x,y\in P$ such that $y\in\overline{P}.$ Then

$\sum_{\bar{z}=y}\mu_{P}(x,z)=\begin{cases}\mu_{\bar{P}}(x,y){\text{ if
}}x=\bar{x}\\\ 0{\text{ otherwise.}}\end{cases}$

In [7], P. Hall proved that if $H<G$, then $\mu(H,G)\neq 0$ only if $H$ is an
intersection of maximal subgroups of $G$. Using the previous theorem, the
following more general statement can be obtained.

######  Proposition 9.

If $H<G$ and $\mu_{A}(H,G)\neq 0,$ then $H$ can be obtained as intersection of
maximal subgroups of $G.$

###### Proof.

Let $H$ be a proper subgroup of $G$ and let $\overline{H}$ be the intersection
of the maximal subgroups of $G$ containing $H$. The map
$[H]_{A}\mapsto[\overline{H}]_{A}$ is a well defined closure map on
$\mathcal{C}_{A}(G),$ so the conclusion follows immediately from Theorem 8. ∎

An element $a$ of a poset $\mathcal{P}$ is called conjunctive if the pair
$\\{a,x\\}$ has a least upper bound, written $a\vee x,$ for each
$x\in\mathcal{P}.$

######  Lemma 10.

[8, Lemma 2.7] Let $\mathcal{P}$ be a poset with a least element $0,$ and let
$a>0$ be a conjunctive element of $\mathcal{P}.$ Then, for each $b>a,$ we have

$\sum_{a\vee x=b}\mu_{\mathcal{P}}(x)=0.$

From the above Lemma 10, it follows easily the following Lemma 11 which,
together with Lemma 12 and Lemma 14 allow us to prove Proposition 1

######  Lemma 11.

Let $N$ be an $A$-invariant normal subgroup of $G$ and $H\leq G.$ If $H<HN<G,$
then

$\mu_{A}(H,G)=-\sum_{[Y]_{A}\in\mathcal{S}_{A}(H,N)}\mu_{A}(H,Y),$

with
$\mathcal{S}_{A}(H,N)=\\{[Y]_{A}\in\mathcal{C}_{A}(G)\mid[H]_{A}\leq[Y]_{A}<[G]_{A}\text{
and }YN=G\\}$.

###### Proof.

Let $\mathcal{P}$ be the interval
$\\{[K]_{A}\in\mathcal{C}_{G}(A)\mid[H]_{A}\leq[K]_{A}\leq[G]_{A}]\\}.$ Notice
that $[HN]_{A}$ is a conjunctive element of $\mathcal{P}.$ Indeed
$[HN]_{A}\vee[K]_{A}=[KN]_{A}$ for every $[K]_{A}\in\mathcal{P}.$ So the
conclusion follows immediately from Lemma 10 ∎

######  Lemma 12.

Let $K$ and $A$ be a subgroup of $G$ and the subgroup of
$\operatorname{Inn}(G)$ induced by the conjugation with the elements of $K$,
respectively. Assume that $N$ is an abelian minimal normal subgroup of $G$
contained in $K$ and $H<HN\leq G.$ Then

$\mu_{A}(H,G)=-\mu_{A}(HN,G)\gamma_{A}(N,H),$

where $\gamma_{A}(N,H)$ is the number of $A$-conjugacy classes of complements
of $N$ in $G$ containing $H.$

###### Proof.

If $HN=G,$ then $H$ is a maximal subgroup of $G,$ hence $\mu_{A}(H,G)=-1,$
while $\mu_{A}(HN,G)=\mu_{A}(G,G)=1$ and $\gamma_{A}(N,H)=1$, so the statement
is true. So we may assume $HN<G$ and apply Lemma 11. Suppose
$[Y]_{A}\in\mathcal{S}_{A}(H,N).$ Let

$\displaystyle\mathcal{C}$ $\displaystyle=\\{J\leq G\mid H\leq J\leq
Y\\},\quad\mathcal{D}=\\{L\leq G\mid HN\leq L\\}$
$\displaystyle\mathcal{C}_{A}$
$\displaystyle=\\{[J]_{A}\in\mathcal{C}_{A}(G)\mid[H]_{A}\leq[J]_{A}\leq[Y]_{A}\\},\
\mathcal{D}_{A}=\\{[L]_{A}\in\mathcal{C}_{A}(G)\mid[HN]_{A}\leq[L]_{A}\\}.$

The map $\eta:\mathcal{C}\to\mathcal{D}$ sending $J$ to $JN$ is an order
preserving bijection. Clearly, if $J_{2}=J_{1}^{x}$ for some $x\in K,$ then
$\gamma(J_{2})=NJ_{2}=NJ_{1}^{x}=(NJ_{1})^{x}=(\gamma(J_{1}))^{x}.$ Conversely
assume $\gamma(J_{2})=(\gamma(J_{1}))^{x}$ with $x\in K.$ Since $YN=G,$ $x=yn$
with $n\in N$ and $y\in Y\cap K.$ Thus $J_{2}N=(J_{1}N)^{x}=(J_{1}N)^{y}$ and
consequently $J_{2}=J_{2}N\cap Y=(J_{1}N)^{y}\cap Y=(J_{1}N\cap
Y)^{y}=J_{1}^{y}.$ It follows that $\eta$ induces an order preserving
bijection from $\mathcal{C}_{A}$ to $\mathcal{D}_{A},$ but then
$\mu_{A}(H,Y)=\mu_{A}(HN,YN)=\mu_{A}(HN,G).$ ∎

The statement of the previous lemma leads to the following open question.

######  Question 13.

Let $G$ be a finite group, $A\leq\operatorname{Aut}(G)$ and $N$ an
$A$-invariant normal subgroup of $G$. Does $\mu_{A}(HN,G)$ divide
$\mu_{A}(H,G)$ for every $H\leq G?$

The following lemma is straightforward.

######  Lemma 14.

Let $A$ be a subgroup of $\operatorname{Aut}(G)$ and $N$ an $A$-invariant
normal subgroup of $G$. Every $a\in A$ induces an automorphism $\overline{a}$
of $G/N$. Let $\overline{A}=\\{\overline{a}\mid a\in A\\}.$ Then, for any
$H\leq G,$ $\mu_{A}(HN,G)=\mu_{\overline{A}}(HN/N,G/N).$

###### Proof of Proposition 1.

We work by induction on $|G|\cdot|G:H|$. The statement is true if $G$ is
abelian. Assume that $G^{\prime}$ contains a minimal normal subgroup, say $N,$
of $G.$ If $N\leq H,$ then, by Lemma 14

$\lambda(H,G)=\lambda(H/N,G/N)=\mu_{\overline{A}}(H/N,G/N)=\mu_{A}(H,G).$

So we may assume $N\not\leq H.$ If $H$ is not an intersection of maximal
subgroups of $G,$ then $\lambda(H,G)=\mu_{A}(H,G)=0$. Suppose
$H=M_{1}\cap\dots\cap M_{t}$ where $M_{1},\dots,M_{t}$ are maximal subgroups
of $G.$ In particular $N$ is not contained in $M_{i}$ for some $i,$ so $M_{i}$
is a complement of $N$ in $G$ containing $H$ and $N\cap H=1.$ By Lemma 12, we
have

$\lambda(H,G)=-\lambda(HN,G)\gamma(N,H),\quad\mu_{A}(H,G)=-\mu_{A}(HN,G)\gamma_{A}(N,H),$

where $\gamma(N,H)$ is the number of conjugacy classes of complements of $N$
in $G$ containing $H$ and $\gamma_{A}(N,H)$ is the number of $A$-conjugacy
classes of these complements. Suppose that $K_{1},K_{2}$ are two conjugated
complements if $N$ in $G$ containing $H.$ Then $K_{2}=K_{1}^{n}$ for some
$n\in N_{N}(H).$ Since $N\leq G^{\prime}\leq K,$ it follows
$\gamma(N,H)=\gamma_{A}(N,H).$ Moreover, by induction,
$\lambda(HN,G)=\mu_{A}(HN,G),$ hence we conclude $\lambda(H,G)=\mu_{A}(H,G).$
∎

##  3\. Generalizing a formula of Philip Hall

We begin with introducing the functions $\Psi_{A}(H,t)$ and $\psi_{A}(H,t)$,
analogue of $\Phi(H,t)$ and $\phi(H,t)$ in the general case of any possible
subgroup $A$ of $\operatorname{Aut}(G)$.

For any $H\in\mathcal{C}_{A}(G)$ and any positive integer $t,$ let

  1. (1)

$\Omega_{A}(H,t)=\bigcup_{a\in A}(H^{a})^{t};$

  2. (2)

$\omega_{A}(H,t)=|\Omega_{A}(H,t)|;$

  3. (3)

$\Psi_{A}(H,t)=\\{(g_{1},\dots,g_{t})\in G^{t}\mid\langle
g_{1},\dots,g_{t}\rangle=H^{a}\text{ for some }a\in A\\};$

  4. (4)

$\psi_{A}(H,t)=|\Psi_{A}(H,t)|.$

If $(x_{1},\dots,x_{t})\in\Omega_{A}(H,t),$ then $\langle
x_{1},\dots,x_{t}\rangle\leq H^{a}$ for some $a\in A,$ hence $\langle
x_{1},\dots,x_{t}\rangle=K$ for some $K\leq G$ with $[K]_{A}\leq[H]_{A}.$ Thus

$\sum_{[K]\leq_{A}[H]}\psi_{A}(K,t)=\omega_{A}(H,t)$

and therefore, by the Möbius inversion formula,

$\sum_{[H]\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)\omega_{A}(H,t)=\psi_{A}(G,t).$

On the other hand $\psi_{A}(G,t)=\phi(G,t)$ so we have proved the following
formula.

######  Theorem 15.

For any finite group $G$ and any subgroup $A$ of $\operatorname{Aut}(G),$

$\phi(G,t)=\sum_{[H]\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)\omega_{A}(H,t).$

Notice that if $A=1,$ then $\omega_{A}(H,t)=|H^{t}|$, so that the result by
Hall given in (1.1) is a particular case of the previous theorem.

######  Corollary 16.

If $G$ is not cyclic, then

$0=\phi(G,1)=\sum_{[H]\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)\omega_{A}(H,1).$

Taking $A=\operatorname{Inn}(G),$ we deduce in particular that if $G$ is not
cyclic, then

$\sum_{H\in\mathcal{C}(H)}\lambda(H,G)\omega_{\operatorname{Inn}(G)}(H,1)=\sum_{H\in\mathcal{C}(H)}\lambda(H,G)|\cup_{g}H^{g}|=0.$

For example, if $G=S_{4},$ then the values of $\lambda(H,G)$ and
$|\cup_{g}H^{g}|$ are as in the following table and $24-12-16-15+4+9+7-1=0.$

| $\lambda(H,G)$ | $|\cup_{g}H^{g}|$
---|---|---
$S_{4}$ | 1 | 24
$A_{4}$ | -1 | 12
$D_{4}$ | -1 | 16
$S_{3}$ | -1 | 15
$K$ | 1 | 4
$\langle(1,2,3,4)\rangle$ | 0 | 10
$\langle(1,2,3)\rangle$ | 1 | 9
$\langle(1,2\rangle$ | 1 | 7
$\langle(1,2)(3,4)\rangle$ | 0 | 4
1 | -1 | 1

If $G=A_{5},$ then the values of $\lambda(H,G)$,
$\omega_{\operatorname{Inn}(G)}(H,1)=|\cup_{g}H^{g}|$,
$\omega_{\operatorname{Inn}(G)}(H,2)=|\cup_{g}(H^{g})^{2}|$ (taking only the
subgroups $H$ with $\lambda(H,G)\neq 0$) are as in the following table and
60-36-36-40+21+32-1=0.

| $\lambda(H,G)$ | $|\cup_{g}H^{g}|$ | $|\cup_{g}(H^{g})^{2}|$
---|---|---|---
$A_{5}$ | 1 | 60 | 3600
$A_{4}$ | -1 | 36 | 636
$S_{3}$ | -1 | 36 | 306
$D_{5}$ | -1 | 40 | 550
$\langle(1,2,3)\rangle$ | 1 | 21 | 81
$\langle(1,2)(3,4)\rangle$ | 2 | 16 | 46
1 | -1 | 1 | 1

Moreover

$3600-636-306-550+81+2\cdot 46-1=2280=\frac{19}{30}\cdot 3600=\phi(A_{5},2).$

If $G=D_{p}=\langle a,b\mid a^{p}=1,b^{2}=1,a^{b}=a^{-1}\rangle$ and $p$ is an
odd prime, then the behaviour of the subgroups in $\mathcal{C}(G)$ is
described by the following table.

| $\lambda(H,G)$ | $|\cup_{g}H^{g}|$
---|---|---
$D_{p}$ | 1 | $2p$
$\langle a\rangle$ | -1 | $p$
$\langle b\rangle$ | -1 | $p+1$
1 | -1 | 1

Another interesting example is given by considering $G=C_{p}^{n}$ and
$A=\operatorname{Aut}(G).$ Let $H\cong C_{p}^{n-1}$ be a maximal subgroup of
$G.$ Then, for $K\leq G$, $\mu_{A}(K,G)\neq 0$ if and only if either
$[K]_{A}=[G]_{A}$ or $[K]_{A}=[H]_{A}.$ Clearly
$\cup_{\alpha\in\operatorname{Aut}(G)}H^{\alpha}=G$ so
$\mu_{A}(G,G)\omega_{A}(G,1)-\mu_{A}(H,G)\omega_{A}(H,1)=|G|-|G|=0.$ More in
general $\Omega_{A}(H,t)$ is the set of $t$-tuples $(x_{1},\dots,x_{t})$ such
that $(x_{1},\dots,x_{t})\in K^{t}$ for some maximal subgroup $K$ of $G$, so
${\mu_{A}(G,G)}\omega_{A}(G,t)-{\mu_{A}(H,G)}\omega_{A}(H,t)=|G|^{t}-\omega_{A}(H,t)$
is the number of generating $t$-tuples of $G.$

Another generalization of (1.1), essentially due to Gaschütz, has been
described by Brown in [1, Section 2.2]. Let $N$ be a normal subgroup of $G$
and suppose that $G/N$ admits $t$ generators for some integer $t$. Let
$y=(y_{1},\dots,y_{t})$ be a generating $t$-tuple of $G/N$ and denote by
$P(G,N,t)$ the probability that a random lift of $y$ to a $t$-tuple of $G$
generates $G$. Then $P(G,N,t)=\phi(G,N,t)|N|^{t}$, where $\phi(G,N,t)$ is the
number of generating $t$-tuples of $G$ lying over $y$. Using again the Möbius
inversion formula it can be proved:

(3.1) $\phi(G,N,t)=\sum_{H\leq G,HN=G}\mu(H,G)|H\cap N|^{t}.$

This formula can be generalized in our contest in the following way:

######  Theorem 17.

Let $N$ be an $A$-invariant normal subgroup of $G$ and fix
$g_{1},\dots,g_{t}\in G$ with the property that $G=\langle
g_{1},\dots,g_{t}\rangle N.$ Define

  * •

$\Omega_{A}(H,N,t)=\\{(n_{1},\dots,n_{t})\mid\langle
g_{1}n_{1},\dots,g_{t}n_{t}\rangle\leq H^{a}\text{ for some }a\in A\\};$

  * •

$\omega_{A}(H,N)=|\Omega_{A}(H,N,t)|$

and let $\mathcal{C}_{A}(G,N)=\\{[H]_{A}\in\mathcal{C}_{A}(G)\mid HN=G\\}.$
Then

$\phi(G,N,t)=\sum_{[H]_{A}\in\mathcal{C}_{A}(G,N)}\mu_{A}(H,G)\omega_{A}(H,N,t).$

###### Proof.

Fix $g_{1},\dots,g_{t}\in G$ with the property that $G=\langle
g_{1},\dots,g_{t}\rangle N.$ Then $\phi(G,N,t)$ is the cardinality of the set

$\Phi(G,N,g_{1},\dots,g_{t})=\\{(n_{1},\dots,n_{t})\in N^{t}\mid\langle
g_{1}n_{1},\dots,g_{t}n_{t}\rangle=G\\}.$

Set:

$\displaystyle\Psi_{A}(H,N,g_{1},\dots,g_{t})=\\{(n_{1},\dots,n_{t})\\!\in\\!N^{t}\mid\langle
g_{1}n_{1},\dots,g_{t}n_{t}\rangle\\!=\\!H^{a}\text{ for some }a\in A\\};$
$\displaystyle\psi_{A}(H,N,t)=|\Psi_{A}(H,N,g_{1},\dots,g_{t})|.$

Notice that $\omega_{A}(H,N,t)\neq 0$ if and only if
$[H]_{A}\in\mathcal{C}_{A}(G,N).$ If
$(n_{1},\dots,n_{t})\in\Omega_{A}(H,N,t),$ then $\langle
g_{1}n_{1},\dots,g_{t}n_{t}\rangle\leq H^{a}$ for some $a\in A,$ and $\langle
g_{1}n_{1},\dots,g_{t}n_{t}\rangle$ $=K$ for some $K\leq G$ with
$[K]_{A}\leq[H]_{A}.$ Thus

$\sum_{[K]_{A}\leq[H]_{A}}\psi_{A}(K,N,t)=\omega_{A}(H,N,t)$

and therefore, by the Möbius inversion formula

$\sum_{[H]\in\mathcal{C}_{A}(G,N)}\mu_{A}(H,G)\omega_{A}(H,N,t)=\psi_{A}(G,N,t)=\phi(G,N,t)\qed$

##  4\. Another application of Möbius inversion formula

Denote by $\Phi^{*}(G,t)$ the set of $t$-tuples $(H_{1},\dots,H_{t})$ of
subgroups of $G$ such that $G=\langle H_{1},\dots,H_{t}\rangle$ and by
$\phi^{*}(G,t)$ the cardinality of this set. For any $H\in\mathcal{C}_{A}(G)$
and any positive integer $t,$ let

  1. (1)

$\Sigma_{A}(H,t)=\\{(H_{1},\dots,H_{t})\mid\langle
H_{1},\dots,H_{t}\rangle\leq H^{a}\text{ for some }a\in A\\};$

  2. (2)

$\sigma_{A}(H,t)=|\Sigma_{A}(H,t)|;$

  3. (3)

$\Gamma_{A}(H,t)=\\{(H_{1},\dots,H_{t})\mid\langle
H_{1},\dots,H_{t}\rangle=H^{a}\text{ for some }a\in A\\};$

  4. (4)

$\gamma_{A}(H,t)=|\Gamma_{A}(H,t)|.$

######  Theorem 18.

$\phi^{*}(G,t)=\sum_{[H]\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)\sigma_{A}(H,t).$

###### Proof.

If $(H_{1},\dots,H_{t})\in\Sigma_{A}(H,t),$ then $\langle
H_{1},\dots,H_{t}\rangle=K$ for some $K\leq G$ with $[K]_{A}\leq[H]_{A}.$ Thus

$\sum_{[K]\leq_{A}[H]}\gamma_{A}(K,t)=\sigma_{A}(H,t)$

and therefore, by the Möbius inversion formula,

$\sum_{[H]\in\mathcal{C}_{A}(G)}\mu_{A}(H,G)\sigma_{A}(H,t)=\gamma_{A}(G,t)=\phi^{*}(G,t).\qed$

In the particular case when $A=1,$ $\sigma_{A}(H,t)=\sigma(H)^{t},$ denoting
with $\sigma(H)$ the number of subgroups of $H.$ So we obtain the following
corollary:

######  Corollary 19.

$\phi^{*}(G,t)=\sum_{H\leq G}\mu(H,G)\sigma(H)^{t}.$

Clearly $\Sigma^{*}(G,t)=\\{G\\},$ so $\phi^{*}(G,1)=1$ and therefore it
follows:

######  Corollary 20.

$1=\sum_{H\in\mathcal{\mathcal{missing}}H_{A}}\mu_{A}(H,G)\sigma_{A}(H,1).$

In particular:

######  Corollary 21.

$1=\sum_{H\leq G}\mu(H,G)\sigma(H).$

For example, if $G=A_{5}$ then the subgroups of $G$ with $\mu(H,G)\neq 0$ are
listed in the following table (where $\kappa(H,G)$ denote the numbers of
conjugate of $H$ in $G$).

| $\mu(H,G)$ | $\kappa(H,G)$ | $\sigma(H)$
---|---|---|---
$A_{5}$ | 1 | 1 | 59
$A_{4}$ | -1 | 5 | 10
$S_{3}$ | -1 | 10 | 6
$D_{5}$ | -1 | 6 | 8
$\langle(1,2,3)\rangle$ | 2 | 10 | 2
$\langle(1,2)(3,4)\rangle$ | 4 | 15 | 2
1 | -60 | 1 | 1

According with Corollary 21, $1=59-5\cdot 10-10\cdot 6-6\cdot 8+2\cdot 10\cdot
2+4\cdot 15\cdot 2-60.$

For a finite group $G$, denote by $P(G,t)$ and $P^{*}(G,t)$ the probability of
generating $G$ with, respectively, $t$ elements or $t$ subgroups. It can be
easily seen that $P(G,t)=P(G/\operatorname{Frat}(G),t),$ but in general
$P^{*}(G,t)\neq P^{*}(G/\operatorname{Frat}(G),t).$ For example, if $G\cong
C_{p^{a}}$, then $G$ and $H\cong C_{p^{a-1}}$ are the unique subgroups of $G$
with non trivial Möbius number and therefore

$\displaystyle P(G,t)$
$\displaystyle=\frac{|G|^{t}-|H|^{t}}{|G|^{t}}=1-\frac{1}{p^{t}},$
$\displaystyle P^{*}(G,t)$
$\displaystyle=\frac{\sigma(G)^{t}-\sigma(H)^{t}}{\sigma(G)^{t}}=1-\frac{a^{t}}{(a+1)^{t}}.$

So $P(G,t)$ is independent on $a$, while $P^{*}(G,t)$ tends to 0 when $a$
tends to infinity.

##  5\. The $(\mu,\lambda)$-property

###### Proof of Thereom 5.

Working by induction on the order of $G,$ it suffices to prove the statement
in the particular case when $N$ is an abelian minimal normal subgroup of $G.$
Let $H$ be a subgroup of $G.$ If $N\leq H,$ then

$\displaystyle\mu(H,G)$
$\displaystyle=\mu(H/N,G/N)=\lambda(H/N,G/N)|N_{G^{\prime}N/N}(H/N):H/N\cap
G^{\prime}N/N|$ $\displaystyle=\lambda(H,G)|N_{G^{\prime}N}(H):H\cap
G^{\prime}N|=\lambda(H,G)|NN_{G^{\prime}}(H):N(H\cap G^{\prime})|$
$\displaystyle=\lambda(H,G)\frac{|N_{G^{\prime}}(H):H\cap G^{\prime}|}{|N\cap
N_{G^{\prime}}(H):N\cap H\cap
G^{\prime}|}=\lambda(H,G)\frac{|N_{G^{\prime}}(H):H\cap G^{\prime}|}{|N\cap
G^{\prime}:N\cap G^{\prime}|}$
$\displaystyle=\lambda(H,G){|N_{G^{\prime}}(H):H\cap G^{\prime}|}.$

So we may assume $N\not\leq H.$ If $H$ is not an intersection of maximal
subgroups of $G,$ then $\mu(G,H)=\lambda(G,H)=0$. So we may assume
$H=M_{1}\cap\dots\cap M_{t}$ where $M_{1},\dots,M_{t}$ are maximal subgroups
of $G.$ Since $N$ is not contained in $H,$ then $N$ is not contained in
$M_{i}$ for some $i,$ but then $M_{i}$ is a complement of $N$ in $G$
containing $H$ and $N\cap H=1.$ If $g\in N_{G}(HN),$ then $g=xn$ with $x\in
M_{i}$ and $n\in N.$ In particular $H^{x}\leq HN\cap M_{i}=H(N\cap M_{i})=H$,
so $N_{G}(HN)=N_{G}(H)N.$ By Lemma 12, we have

$\frac{\mu(H,G)}{\lambda(H,G)}=\frac{\mu(HN,G)}{\lambda(HN,G)}\frac{\kappa}{\delta}=|N_{G^{\prime}N}(HN):HN\cap
G^{\prime}N|\frac{\kappa}{\delta}=|NN_{G^{\prime}}(H):HN\cap
G^{\prime}N|\frac{\kappa}{\delta}$

where $k$ is the number of complements of $N$ in $G$ containing $H$ and
$\delta$ is the number of conjugacy classes of these complements. First assume
that $N\leq Z(G).$ Then $\kappa=\delta$, $G^{\prime}=M_{i}^{\prime}\leq
M_{i},$ $N\cap G^{\prime}=1$ and

$\displaystyle\frac{\mu(H,G)}{\lambda(H,G)}$
$\displaystyle=|NN_{G^{\prime}}(H):HN\cap
G^{\prime}N|\frac{\kappa}{\delta}=|NN_{G^{\prime}}(H):HN\cap G^{\prime}N|$
$\displaystyle=|NN_{G^{\prime}}(H):N(H\cap
G^{\prime})|=|N_{G^{\prime}}(H):H\cap G^{\prime}|.$

Finally assume $N\not\leq Z(G).$ Then $N\leq G^{\prime}$,
$\kappa/\delta=|N_{N}(H)|$ and

$\displaystyle\frac{\mu(H,G)}{\lambda(H,G)}$
$\displaystyle=|NN_{G^{\prime}}(H):HN\cap
G^{\prime}N|\frac{\kappa}{\delta}=|NN_{G^{\prime}}(H):N(H\cap
G^{\prime})||N_{N}(H)|$
$\displaystyle=\frac{|N||N_{G^{\prime}}(H)|}{|N_{N}(H)|}\frac{|N_{N}(H)|}{|N||H\cap
G^{\prime}|}=|N_{G^{\prime}}(H):H\cap G^{\prime}|.\qed$

###### Proof of Corollary 6.

Suppose that $G$ has minimal order with respect to the property that $G$ does
not satisfy the $(\mu,\lambda)$ property. By the previous proposition, $G$
contains no abelian minimal normal subgroup and therefore
$\operatorname{soc}(G)=S_{1}\times\dots\times S_{t}$ is a direct product of
nonabelian finite simple groups. If $|G|\leq|PSU(3,3)|=6048$, then either
$t=1$ or $G=\operatorname{soc}(G)=A_{5}\times A_{5}.$ So it suffices to check
that $A_{5}\times A_{5}$ and any almost simple group of order at most 6048
satisfies the $(\mu,\lambda)$ property. Since, for every $H\leq G,$
$\lambda(H,G)$ and $\mu(H,G)$ can be computed from the table of marks of $G$
(see [10, Proposition 1]), this task can be easely completed using the library
of table of marks available in GAP [4]. ∎

We may use Theorem 15 to deduce some consequences of the
$(\mu,\lambda)$-property.

######  Theorem 22.

Suppose that a finite group $G$ satisfies the $(\mu,\lambda)$-property. Then

(5.1)
$\sum_{[H]\in\mathcal{C}(G)}\lambda(H,G)\left(\frac{|H|^{t-1}|G||G^{\prime}H|}{|G^{\prime}N_{G}(H)|}-\omega(H,t)\right)=0.$

###### Proof.

By Theorem 15,

$\displaystyle\sum_{H\in\mathcal{C}(G)}\lambda(H,G)\omega(H,t)$
$\displaystyle=\phi(G,t)=\sum_{H\leq G}\mu(H,G)|H|^{t}$
$\displaystyle=\sum_{H\in\mathcal{C}(G)}\mu(H,G)|G:N_{G}(H)||H^{t}|$
$\displaystyle=\sum_{H\in\mathcal{C}(G)}\lambda(H,G)|N_{G^{\prime}}(H):G^{\prime}\cap
H||G:N_{G}(H)||H^{t}|$
$\displaystyle=\sum_{H\in\mathcal{C}(G)}\lambda(H,G)\frac{|H|^{t}|G||N_{G^{\prime}}(H)|}{|G^{\prime}\cap
H||N_{G}(H)|}$
$\displaystyle=\sum_{H\in\mathcal{C}(G)}\lambda(H,G)\frac{|H|^{t-1}|G||G^{\prime}H|}{|G^{\prime}N_{G}(H)|}.\qed$

A natural question is whether (5.1) is also a sufficient condition for the
$(\mu,\lambda)$-property. For any $H\leq G,$ set
$\mu^{*}(H,G)=|N_{G^{\prime}}(H):G^{\prime}\cap H|\lambda(H,G)$. The validity
of $(\ref{strana})$ is equivalent to

$\sum_{H\in\mathcal{C}(G)}\lambda(H,G)\omega(H,t)-\sum_{H\in\mathcal{C}(G)}\mu^{*}(H,G)|H|^{t}|G:N_{G}(H)|=0.$

In any case we must have

$\sum_{H\in\mathcal{C}(G)}\lambda(H,G)\omega(H,t)-\sum_{H\in\mathcal{C}(G)}\mu(H,G)|H|^{t}|G:N_{G}(H)|=0.$

So $(\ref{strana})$ is equivalent to

$\sum_{H\in\mathcal{C}(G)}\frac{(\mu(H,G)-\mu^{*}(H,G))|H|^{t}}{|N_{G}(H)|}=0.$

Let $\mathcal{T}=\\{[H]\in\mathcal{C}(G)\mid\mu(H,G)\neq\mu^{*}(H,G)\\}.$ Then
$(\ref{strana})$ is true if and only if

(5.2)
$\sum_{[H]\in\mathcal{T}}\frac{(\mu(H,G)-\mu^{*}(H,G))|H|^{t}}{|N_{G}(H)|}=0.$

For example, if $G=PSU(3,3),$ then $\mathcal{T}$ consists of four conjugacy
classes of subgroups and the corresponding values are given by the following
table:

$\mu(H,G)$ | $\mu^{*}(H,G)$ | $|H|$ | $|N_{G}(H)|$
---|---|---|---
-48 | 0 | 2 | 96
3 | 0 | 6 | 18
0 | -4 | 8 | 32
1 | 2 | 24 | 24

In this case (5.2) is equivalent to

$2^{t-1}-6^{t-1}-8^{t-1}+24^{t-1}=0$

which is true only if $t=1.$

For any positive integer $n$ let

$\tau(n)=\sum_{H\in\mathcal{T},|H|=n}\frac{\mu(H,g)-\mu^{*}(H,G)}{|N_{G}(H)|}.$

######  Proposition 23.

A finite group $G$ satisfies (5.1) for any positive integer $t$ if and only if
$\tau(n)=0$ for any $\in\mathbb{N}.$

######  Question 24.

Does $\tau(n)=0$ for any $n\in\mathbb{N}$ implies $\mu^{*}(H,G)=\mu(H,G)$ for
any $H\leq G?$

For any $H\leq G,$ consider

$\alpha(H,t)=\frac{|H|^{t-1}|G||G^{\prime}H|}{|G^{\prime}N_{G}(H)|},\quad\beta(H,t)=\alpha(H,t)-\omega(H,t).$

Let $\mathcal{C}^{*}(G)=\\{[H]\in\mathcal{C}(H)\mid[H]<[G]\text{ and
}\lambda(H,G)\neq 0\\}.$ If $G$ satisfies the $(\lambda,\mu)$-property, then
for any $t\in\mathbb{N},$ the vector

$\beta_{t}(G)=(\beta(H,t))_{[H]\in\mathcal{C}^{*}(G)}$

is an integer solution of the linear equation

(5.3) $\sum_{\mathcal{[}H]\in C^{*}(G)}\lambda(H,G)x_{H}=0.$

One could investigate about the dimension of the vector space generate by the
vectors $\beta_{t}(G),$ $t\in\mathbb{N}.$ For example, if $G=A_{5}$, then we
may order the elements of $\mathcal{C}^{*}(G)$ so that $H_{1}=A_{4},$
$H_{2}=S_{3},$ $H_{3}=D_{5},$ $H_{4}=\langle(1,2,3)\rangle,$
$H_{5}=\langle(1,2)(3,4)\rangle,$ $H_{6}=1.$ Then (5.3) can be written in the
form

$\sum_{[H]\in\mathcal{C}^{*}(G)}\lambda(H,G)x_{H}=-x_{H_{1}}-x_{H_{2}}-x_{H_{3}}+x_{H_{4}}+2x_{H_{5}}-x_{H_{6}}$

and

$\displaystyle\beta_{1}(G)$ $\displaystyle=(24,24,20,39,44,59),$
$\displaystyle\beta_{2}(G)$ $\displaystyle=(84,54,50,99,74,59),$
$\displaystyle\beta_{3}(G)$ $\displaystyle=(264,114,110,279,134,59),$
$\displaystyle\beta_{4}(G)$ $\displaystyle=(804,234,230,819,254,59),$
$\displaystyle\beta_{5}(G)$ $\displaystyle=(2424,474,470,2439,494,59),$
$\displaystyle\beta_{6}(G)$ $\displaystyle=(7284,954,950,7299,974,59).$

The first three vectors $\beta_{1}(G),\beta_{2}(G),\beta_{3}(G)$ are linearly
independent, while $\beta_{4}(G),$ $\beta_{5}(G)$ and $\beta_{6}(G)$ can be
obtained as linear combinations of $\beta_{1}(G),\beta_{2}(G),\beta_{3}(G).$

The situation is completely different when $G=S_{3}$. We may order the
elements of $C^{*}(G)$ so that $H_{1}=\langle(1,2,3)\rangle,$
$H_{2}=\langle(1,2)\rangle,$ $H_{3}=1.$ The equation (5.3) has in this case
the form $x_{H_{1}}+x_{H_{2}}-x_{H_{3}}=0$ and $\beta_{t}(G)=(0,2,2)$
independently on the choice of $t.$

Some properties of the vectors $\beta_{t}(G)$ are described in the following
propositions.

######  Proposition 25.

If $H\in\mathcal{C}^{*}(G),$ then $\beta(H,t)\geq 0$ with equality if and only
if $G^{\prime}\leq H.$ In particular $\beta_{t}(G)$ is a non-negative vector
and $\beta_{t}(G)=0$ if and only if $G$ is nilpotent.

###### Proof.

Notice that $\omega(H,t)\leq|G:N_{G}(H)|(|H|^{t}-1)+1.$ So

$\displaystyle\beta(H,t)$
$\displaystyle\geq\frac{|H|^{t-1}|G||G^{\prime}H|}{|G^{\prime}N_{G}(H)|}-|G:N_{G}(H)|(|H|^{t}-1)-1$
$\displaystyle=|H|^{t}|G:N_{G}(H)|\frac{|G^{\prime}\cap
N_{G}(H)|}{|G^{\prime}\cap H|}-|G:N_{G}(H)|(|H|^{t}-1)-1\geq 0$

with equality if and only if $H\geq G^{\prime}.$ ∎

######  Proposition 26.

The vector $\beta_{t}(G)$ is independent on the choice of $t$ if and only if
$G$ is a nilpotent group or a primitive Frobenius group, with cyclic Frobenius
complement.

###### Proof.

By the previous proposition, if $G$ is nilpotent then $\beta_{t}(G)$ is the
zero vector for any $t\in\mathbb{N},$ so we may assume that $G$ is not
nilpotent. Assume that $\beta_{t}(G)$ is independent on the choice of $t.$ Let
$H$ be a maximal non-normal subgroup of $G$. Then $\alpha(H,t)=|H|^{t}\cdot u$
with $u=|G:H|$. Let $H_{1},\dots,H_{u}$ be the conjugates of $H$ in $G.$ For
any $J\subseteq\\{1,\dots,u\\},$ let $\alpha_{J}=|\cap_{j\in J}H_{j}|.$ Then

$\beta(H,t)=\sum_{J\neq\\{1,\dots,u\\}}(-1)^{|J|+1}|\alpha_{J}|^{t}.$

We must have $\alpha_{J}=1$ for every choice of $J,$ otherwise
$\lim_{t\to\infty}\beta(H,t)=\infty.$ Hence $H$ is a Frobenius complement and,
since $H$ is a maximal subgroup, the Frobenius kernel $V$ is an irreducible
$H$-module. Since $\beta(V,t)=|V|^{t}(|H^{\prime}|-1)$ does not depends on
$t$, $H$ must be abelian, and consequently cyclic. So if $\beta_{t}(G)$ is
independent of the choice of $t,$ then $G$ is a primitive Frobenius group with
a cyclic Frobenius complement. Conversely assume $G=V\rtimes H,$ where $H$ is
cyclic and $V$ and irreducible $H$-module. If $X\in\mathcal{C}^{*}(G)$, then
$\lambda(X,G)\neq 0,$ so $X$ is an intersection of maximal subgroups of $G$
and therefore either $V=G^{\prime}\leq X,$ or $X$ is conjugate to a subgroup
of $H.$ In the first case $\beta(H,t)=0$. Assume $X=K^{v}$ for some $K\leq H$
and $v\in V.$ Then
$\beta(H,t)=|K|^{t}|V|-\omega(K,t)=|K|^{t}|V|-(|V|(|K|^{t}-1)+1)=|V|-1.$ ∎

## References

  * [1] K. S. Brown, The coset poset and probabilistic zeta function of a finite group, J. Algebra 225 (2000), no. 2, 989–1012.
  * [2] H. Crapo, The Möbius function of a lattice, J. Combinatorial Theory 1 (1966), 126–134.
  * [3] H. Crapo, Möbius inversion in lattices, Arch. Math. (Basel) 19 (1968) 595–-607.
  * [4] The GAP Group, _Gap – groups, algorithms, and programming, version 4.11.1_ , 2021, https://www.gap-system.org.
  * [5] W. Gaschütz, Zu einem von B. H. und H. Neumann gestellten Problem, Mathematische Nachrichten 14 (1955), no. 4-6, 249–252.4
  * [6] F. Dalla Volta and G. Zini, On two Möbius functions for a finite non-solvable group, Comm. Algebra 49 (2021), no. 11, 4565–4576.
  * [7] P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134–151.
  * [8] T. Hawkes, I. M. Isaacs and M. Özaydin, On the Möbius function of a finite group. Rocky Mountain J. Math. 19 (1989), no. 4, 1003–1034.
  * [9] A. Lucchini, On the subgroups with non-trivial Möbius number, J. Group Theory 13 (2010), no. 4, 589–600.
  * [10] H. Pahlings, On the Möbius function of a finite group, Arch. Math. (Basel) 60 (1) (1993), 7–14.