#  A torus of $N$-dimensional charged anti-de-Sitter black holes in the
quadratic form of $f(Q)$ gravitational theory

G. G. L. Nashed *  —<EMAIL_ADDRESS>Centre for Theoretical Physics, The
British University, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt

(August 28, 2024)

###### Abstract

Due to the absence of spherically symmetric black hole solutions in
$f(\mathbb{Q})$ because of the constraint derived from its field equations,
which yields either $\mathbb{Q}=0$ or $f_{\mathbb{Q}\mathbb{Q}}=0$ Heisenberg
(2023); Maurya et al. (2023). We are going to introduce a tours solutions for
charged anti-de-Sitter black holes in $N$-dimensions within the framework of
the quadratic form of $f(\mathbb{Q})$ gravity, where the coincident gauge
condition is applied Heisenberg (2023). Here,
$f(\mathbb{Q})=\mathbb{Q}+\frac{1}{2}\alpha\mathbb{Q}^{2}-2\Lambda$, and the
condition $N\geq 4$ is satisfied. These black hole solutions exhibit flat or
cylindrical horizons as their distinctive features. An intriguing aspect of
these black hole solutions lies in the coexistence of electric monopole and
quadrupole components within the potential field, which are indivisible and
exhibit interconnected momenta. This sets them apart from the majority of
known charged solutions in the linear form of the non-metricity theory and its
extensions. Moreover, the curvature singularities in these solutions are less
severe compared to those found in known charged black hole solutions within
the characteristic can be demonstrated by computing certain invariants of the
curvature and non-metricity tensors. Finally, we calculate thermodynamic
parameters, including entropy, Hawking temperature, and Gibbs free energy.
These thermodynamic computations affirm the stability of our model.

Modified gravity; non-metricity theory; black holes; singularities.

###### pacs:

04.50.Kd, 98.80.-k, 04.80.Cc, 95.10.Ce, 96.30.-t

##  I Introduction

While Einstein’s general relativity (GR) has undeniably enjoyed success, its
constraints in various aspects have become evident in recent years, leading
scholars to investigate alternative theories. Subsequently, on flat spacetime,
an affine connection that is compatible with the metric and exhibits torsion
was introduced, supplanting the initially used Levi-Civita connection that is
both free of torsion and compatible with the metric, serving as the foundation
for GR. This allowed the torsion to assume the entire responsibility for
characterizing gravity. The theory introduced by Einstein himself Unzicker and
Case (2005), known as metric teleparallel gravity, is of particular note.
Recently, a newcomer has joined this family, the symmetric teleparallel
theory. It is derived from an affine connection characterized by both
vanishing curvature and torsion, attributing the gravitational effects to the
non-metricity of spacetime Nester and Yo (1999). In the metric teleparallel
theory and its symmetric counterpart, one can build the torsion scalar
$\mathbb{T}$ from the torsion tensor and the non-metricity scalar $\mathbb{Q}$
respectively. Subsequently, by adopting the Lagrangian
$\mathcal{L}=\sqrt{-g}\mathbb{T}$ in the former and
$\mathcal{L}=\sqrt{-g}\mathbb{Q}$ in the latter, the corresponding field
equations can be derived.

Nevertheless, given that the scalars $\mathbb{T}$ and $\mathbb{Q}$ are the
same as the Levi-Civita Ricci scalar, modulo a surface term, both of these
theories are equivalent to GR up to a boundary term. Undoubtedly, the “dark”
issue that metric and symmetric teleparallelism experience in GR is shared by
both. In order to address this difficulty, updated gravity theories have been
presented in their respective fields, with $f(\mathbb{T})$ Ferraro and Fiorini
(2007) and $f(\mathbb{Q})$ Beltrán Jiménez et al. (2018a). It is important to
emphasize that the teleparallel theory incorporates an affine link independent
of the metric tensor, in contrast to GR. Therefore, teleparallel theories can
be thought of as metric-affine theories in which the connection and the metric
are both dynamic variables.

Unlike $f(\mathbb{R})$ which is fourth-order equations, the two theories,
$f(\mathbb{Q})$ and $f(\mathbb{T})$, display second-order field equations.
This directly outperforms $f(\mathbb{R})$ and positions $f(\mathbb{Q})$ and
$f(\mathbb{T})$ as viable explanations for the universe’s accelerating
expansion Bengochea and Ferraro (2009); Linder (2010); Bamba et al. (2011,
2010); Narawade et al. (2022); Solanki et al. (2022a, b); Atayde and
Frusciante (2021). Nevertheless, models grounded in $f(\mathbb{T})$ encounter
significant challenges related to coupling and the local Lorentz invariance
problem, concerns that do not apply to the theories of non-metricity. More
research is warranted given the deeper similarities between these two
intriguing subfields of teleparallel theories of gravity, see Beltrán Jiménez
et al. (2019); D’Ambrosio et al. (2022); Lu et al. (2021); Capozziello et al.
(2022) and the references therein.

The principle of equivalence requires gravity to demonstrate a geometric
essence. Einstein’s GR serves as a gravity theory based on geometry,
portraying a Riemannian manifold as spacetime. In this context, the affine
connection is the Levi-Civita connection, which is both metric-compatible and
torsion-free, and is entirely determined by the metric. In GR, the fundamental
quantity characterizing the manifold is the scalar curvature $\mathbb{R}$.
However, the demand for Riemannian geometry is arbitrary, and typically, a
manifold is defined by three essential geometric attributes: torsion
$\mathbb{T}$, curvature $\mathbb{R}$, and non-metricity $\mathbb{Q}$ Beltrán
Jiménez et al. (2019). As such, many gravitational theories can be developed
through examining the characteristics of their relationship. In metric-affine
geometry, subcategories in particular are identified as possessing Riemann-
Cartan ($\mathbb{Q}=0$), teleparallel ($\mathbb{R}=0$), and no torsion
($\mathbb{T}=0$). Additional subsets arise when $\mathbb{Q}$ and $\mathbb{T}$,
or $\mathbb{Q}$ and $\mathbb{R}$, or $\mathbb{R}$ and $\mathbb{T}$,
simultaneously disappear. Weitzenböck or teleparallel
($\mathbb{R}=\mathbb{Q}=0$), Riemannian $\mathbb{T}=\mathbb{Q}=0$), and
symmetric teleparallel ($\mathbb{R}=\mathbb{T}=0$) are the names given to
these geometries. When all three quantities disappear simultaneously, the
final trivial subset emerges, resulting in a Minkowskian manifold.

It is important to note that there are other formulations of GR that are
equivalent or alternative to it. The expression “teleparallel equivalent of
general relativity” (TEGR) refers to one such formulation Aldrovandi and
Pereira (2013); Maluf (2013); Dialektopoulos et al. (2019); Barros et al.
(2020); Beltrán Jiménez et al. (2020); Bajardi et al. (2020); Ayuso et al.
(2021); Flathmann and Hohmann (2021); Khyllep et al. (2021); D’Ambrosio et al.
(2020). It is distinguished by lacking curvature and non-metricity. The
symmetric teleparallel equivalent of GR (STEGR) Nester and Yo (1999); Adak and
Sert (2005); Adak et al. (2006, 2013); Mol (2017); Beltrán Jiménez et al.
(2018a, b); Gakis et al. (2020) is another formulation that is equivalent,
where torsion and curvature both disappear. Similar to GR, the Lagrangian
density in these analogous theories is aligned with the corresponding scalars
$\mathbb{T}$ and $\mathbb{Q}$. The reference contains up-to-date analysis and
Comparisons between these comparable formulas are available in Järv et al.
(2018); Capozziello et al. (2022); Heisenberg (2019, 2023).

While GR has effectively explained the interplay within the solar system and
expansive cosmic structures, numerous unresolved issues persist. These include
mysteries such as the unknown components of the Universe, early-time inflation
exists, and quantizing it presents certain difficulties. It is now generally
acknowledged that GR (or similar formulations) might not be the final
explanation of gravity, necessitating possible revisions. Therefore, the
simplest straightforward change to deal with GR, the TEGR, or the STEGR is to
alter the Lagrangian density so that it becomes a function of the pertinent
scalars of non-metricity, curvature, or torsion. The strategies that emerge
are known as the theories of $f(\mathbb{R})$-gravity De Felice and Tsujikawa
(2010); Sotiriou and Faraoni (2010) $f(\mathbb{T})$-gravity Cai et al. (2016);
Bahamonde et al. (2023); Bamba et al. (2012, 2013); Casalino et al. (2021) and
$f(\mathbb{Q})$-gravity Beltrán Jiménez et al. (2018a, b); Zhao (2022); Lazkoz
et al. (2019); Mandal et al. (2020, 2020); Capozziello and Shokri (2022);
Capozziello and D’Agostino (2022). Recent analysis have examined the
similarities and differences with respect to $f(\mathbb{R})$, $f(\mathbb{T})$,
and $f(\mathbb{Q})$ gravity in terms of degrees of freedom and symmetry
breaking Hu et al. (2022).

Deriving analytical solutions in theories that incorporate elevated curvature
or torsion is not always a straightforward task, even within the frame of
$f(\mathbb{T})$ Nashed (2013a); Capozziello et al. (2013); Nashed (2013b).
Solutions possessing a cosmological constant demonstrate compelling
characteristics, such as the emergence of diverse horizon topologies, contrary
to the asymptotic flat scenario. Within such instances, the horizons of black
holes can manifest as spherical, hyperbolic, or planar, potentially giving
rise to toroidal or cylindrical formations determined by the chosen global
identifications Mann (1997). The investigation of charged black holes
approaching de Sitter and Anti-de-Sitter spaces, n conjunction with black
holes undergoing rotation in four dimensions or beyond, has been thorough
within the frame of AdS/CFT and dS/CFT correspondences, for further details on
these studies, c.f., Lemos (1995); Awad and Johnson (2000a, b); Aharony et al.
(2000).

Due to the incapability of $f(\mathbb{Q})$ gravity to derive in its frame a
spherically symmetric solution due to the constraint given by its field
equations, $\mathbb{Q}=0$ or $f_{\mathbb{Q}\mathbb{Q}}=0$ Heisenberg (2023);
Maurya et al. (2023). Therefore, within this study, we will introduce novel
solutions for charged black holes asymptotically approaching Anti-de-Sitter
spaces with flat horizons within the context of $f(\mathbb{Q})$ theories.
Here, $f(\mathbb{Q})=\mathbb{Q}+\frac{1}{2}\alpha\mathbb{Q}^{2}$, with
$\alpha<0$, and $N\geq 4$. Primarily, these solutions for black holes feature
electric potentials that include both monopole and quadruple terms. To attain
an asymptotic Anti-de-Sitter (AdS) solution, it becomes necessary to establish
a linkage between the momentum of electric monopole and the momentum of the
quadruple solution. Consequently, these two momenta become indistinguishable.
Another noteworthy aspect is that, despite the singularity at $r=0$, the
singularity of this black hole solution is evidently less severe compared to
the singularities observed in asymptotic AdS charged black hole solutions in
Einstein theory. As an example, in N-dimensions, the Kretschmann invariant,
obtained from squaring of the Riemannian tensor, Ricci tensor and the Ricci
invariant exhibit behaviors such as
$\mathbb{R}_{\mu\nu\alpha\rho}\mathbb{R}^{\mu\nu\alpha\rho}=\mathbb{R}_{\mu\nu}\mathbb{R}^{\mu\nu}=\mathbb{R}=\mathbb{Q}_{\mu\nu\alpha}\mathbb{Q}^{\mu\nu\alpha}=\mathbb{Q}_{\mu}\mathbb{Q}^{\mu}\sim
r^{-(N-2)}$ and $\mathbb{\bar{Q}}_{\mu}\mathbb{\bar{Q}}^{\mu}=\mathbb{Q}\sim
r^{-3(N-2)}$. This is contrary to the familiar solutions of Einstein-Maxwell
and TEGR theories, where Kretschmann and Ricci scalars follow patterns like
$K\sim r^{-3(N-3)}$ and $\mathbb{R}_{\mu\nu}\mathbb{R}^{\mu\nu}\approx\sim
r^{-2N}$ and
$\mathbb{Q}_{\mu\nu\alpha}\mathbb{Q}^{\mu\nu\alpha}=\mathbb{Q}_{\mu}\mathbb{Q}^{\mu}\sim
r^{-(N-1)}$ and $\mathbb{\bar{Q}}_{\mu}\mathbb{\bar{Q}}^{\mu}\sim r^{-2N)}$,
and $\mathbb{Q}\sim r^{-N}$. Moreover, despite variations in the $g_{tt}$ and
$g^{rr}$ components of the metric for these asymptotically Anti-de-Sitter
charged solutions, they share common Killing and event horizons.

This paper is outlined as: In Section II, we provide a concise overview of the
non-mitricity formalism by scrutinizing the field equations. Following that,
we provide the equation of motions for gravity in the context of
$f(\mathbb{Q})$. In Section III, a metric potential field that has a tours in
N-dimensions is utilized to the equations of motion of $f(\mathbb{Q})$ to
provide the derivation of a comprehensive neutral black hole solution in
N-dimensions. The solution exhibits asymptotic behavior consistent with Anti-
de-Sitter (AdS) space. Moving on to Section IV, a cylindrically symmetric
line-element is employed under the $f(\mathbb{Q})$ gravity framework in the
non-metricity-Maxwell field equations. We demonstrate the reduction of this
solution to a precise static black hole with charge in Anti-de-Sitter (AdS)
space. Notably, this black hole exhibits both monopole and quadrupole momenta.
Section V outlines pertinent physical characteristics of these black holes. In
Section VI, we delve into the thermodynamics of the black holes. In Section
VII, we explore black holes with multiple horizons as introduced in Section
IV. Ultimately, concluding remarks are provided in Section VIII.

##  II $f(\mathbb{Q})$-gravitational theory

We provide a summary of some general features of $f(\mathbb{Q})$ gravitational
theory in this section. We’ll restrict our exposition to components, and for a
more comprehensive derivation concerning forms, readers are directed to the
references Banerjee et al. (2021); Lin and Zhai (2021)).

The expression for the general affine connection on a manifold that is both
parallelizable and differentiable is as follows:

$\Gamma^{\sigma}_{\;\mu\nu}=\tilde{\Gamma}^{\sigma}_{\;\mu\nu}+K^{\sigma}_{\;\mu\nu}+L^{\sigma}_{\;\mu\nu}\,.$
(1)

The Levi-Civita connection in this case is
$\tilde{\Gamma}^{\sigma}_{\;\mu\nu}$, and characterized using the metric
as111Within this study, we adopt relativistic units with c = G = 1.
Consequently, the Einstein constant, denoted by $\kappa$, is equivalent to
$8\pi$. The symmetric part will be represented by parentheses, denoted as
$()$. For instance, $A_{(\mu\nu)}$ is defined as
$(1/2)(A_{\mu\nu}+A_{\nu\mu})$. Conversely, the antisymmetric part is denoted
by square brackets, represented as [ ], such that $A_{[\mu\nu]}$ is defined as
$(1/2)(A_{\mu\nu}-A_{\nu\mu})$.:

$\tilde{\Gamma}^{\sigma}_{\;\mu\nu}=\frac{1}{2}g^{\sigma\rho}\left(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\right)\,.$
(2)

Furthermore, $K^{\sigma},_{\mu\nu}$ represents the contortion which is defined
as:

$K^{\sigma}_{\;\mu\nu}=\frac{1}{2}T^{\sigma}_{\;\mu\nu}+T^{\;\;\;\sigma}_{(\mu\;\;\nu)}\;\;\;\;.$
(3)

Here $T^{\sigma}_{\;\mu\nu}=2\Gamma^{\sigma}_{\;[\mu\nu]}$ is the torsion
tensor. Finally, $L^{\sigma}_{;\mu\nu}$ denotes the deformation and is
expressed as follows:

$L^{\sigma}_{\;\mu\nu}=\frac{1}{2}\mathbb{Q}^{\sigma}_{\;\mu\nu}-\mathbb{Q}^{\;\;\;\sigma}_{(\mu\;\;\nu)}\,.$
(4)

Here $\mathbb{Q}^{\sigma}_{\;\mu\nu}$ represents the non-metricity tensor
expressed as:

$\mathbb{Q}_{\sigma\mu\nu}=\nabla_{\sigma}g_{\mu\nu}=\partial_{\sigma}g_{\mu\nu}-\Gamma^{\rho}_{\;\sigma\mu}g_{\nu\rho}-\Gamma^{\rho}_{\;\sigma\nu}g_{\mu\rho}\,.$
(5)

Consequently, the scalar of the non-metricity is defined as:

$\displaystyle\mathbb{Q}=$ $\displaystyle
g^{\mu\nu}(L^{\alpha}_{\beta\nu}L^{\beta}_{\mu\alpha}-L^{\beta}_{\alpha\beta}L^{\alpha}_{\mu\nu})$
$\displaystyle\equiv$
$\displaystyle-\mathbb{Q}_{\sigma\mu\nu}P^{\sigma\mu\nu}\,.$ (6)

Here $P^{\sigma\mu\nu}$ represents the non-metricity conjugate defined as:

$P^{\sigma}_{\;\mu\nu}=\frac{1}{4}\left(-\mathbb{Q}^{\sigma}_{\;\mu\nu}+2Q^{\;\;\;\sigma}_{(\mu\;\;\nu)}+\mathbb{Q}^{\sigma}g_{\mu\nu}-\tilde{\mathbb{Q}}^{\sigma}g_{\mu\nu}-\delta^{\sigma}_{(\mu}\mathbb{Q}_{\nu)}\right)\,.$
(7)

Here $\mathbb{Q}_{\sigma}$ and $\tilde{\mathbb{Q}}_{\sigma}$ are defined as:

$\mathbb{Q}_{\sigma}=\mathbb{Q}^{\;\;\mu}_{\sigma\;{\mathbf{\cdot}}\;\mu}\,\qquad\mbox{and}\quad\tilde{\mathbb{Q}}_{\sigma}=\mathbb{Q}^{\mu}_{\;{\mathbf{\cdot}}\;\sigma\mu}\,.$

In the event when both the torsion and the non-metricity equal zero, the
connection will resemble the Levi-Civita connection. Torsion and curvature
vanish in STEGR gravity, and non-metricity depends on metric and connection.

Modified STEGR was disucssed in Ref. Beltrán Jiménez et al. (2018a) where the
action coupled with Maxwell field yields the form:

$I=-\frac{1}{2\kappa^{2}}\int_{\mathcal{M}}f(\mathbb{Q})\sqrt{-g}d^{N}x+\int\sqrt{-g}{\mathcal{L}}_{em}\leavevmode\nobreak\
d^{N}x\,.$ (8)

Here, $g$ , $g_{\mu\nu}$, and $\mathcal{M}$ represents the determinant, the
covariant metric tensor, and the space-time manifold, respectively. The
function $f(\mathbb{Q})$ represents the general functional form of the non-
metricity scalar $\mathbb{Q}$. The definition of $\kappa$ is given by
$\kappa=2(N-3)\Omega_{N-1}G_{N}$, where $G_{N}$ represents the constant of
gravitational field in $N$-dimension. In this study, $\Omega_{N-1}$ denotes
the $(N-1)$-dimensional unit sphere volume. The term of $\Omega_{N-1}$ is
specified as $\Omega_{N-1}=\frac{2\pi^{(N-1)/2}}{\Gamma((N-1)/2)}$. The
$\Gamma$-function’s dependence is contingent on the dimension of
spacetime222For instance, when $N=4$, it can be demonstrated that
$2(N-3)\Omega_{N-1}=8\pi$.. In Eq. (8),
${\mathcal{L}}_{em}=-\frac{1}{2}{F}\wedge^{\star}{F}$ represents the Maxwell
field Lagrangian, where $F=dA$, and $A=A_{\mu}dx^{\mu}$, is the 1-form of the
electromagnetic potential Awad et al. (2017).

When determining the theory’s field equations, one conducts separate
variations concerning both the metric and the connection with respect to (8)

$\displaystyle\zeta_{\mu\nu}=\frac{2}{\sqrt{-g}}\nabla_{\alpha}\left(\sqrt{-g}f_{\mathbb{Q}}P^{\alpha}_{\;\;\mu\nu}\right)+\frac{1}{2}g_{\mu\nu}f+f_{\mathbb{Q}}\left(P_{\mu\alpha\beta}\mathbb{Q}^{\;\;\alpha\beta}_{\nu}-2P_{\alpha\beta\mu}\mathbb{Q}^{\;\;\alpha\beta}_{\nu}\right)+\kappa^{2}\frac{1}{2}\kappa{{{\mathcal{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}_{\mu\nu}$
$\displaystyle\partial_{\nu}\left(\sqrt{-g}F^{\mu\nu}\right)=0\;,$ (9)

An extra constraint on the connection can be achieved with the aid of Eq. (8),
as demonstrated by

$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}f_{\mathbb{Q}}P^{\alpha}_{\;\;\mu\nu}\right)=0\,.$
(10)

In this study,
${{{\mathcal{T}}^{{}^{{}^{{!!!!\scriptstyle{em}}}}}}}^{\nu}_{\mu}$ is the
tensor representing the electromagnetic field’s energy-momentum, defined as:

${{{\mathcal{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}=F_{\mu\alpha}F^{\nu\alpha}-\frac{1}{4}\delta_{\mu}{}^{\nu}F_{\alpha\beta}F^{\alpha\beta}.$

Here, as is customary, $\mathcal{T}_{\mu\nu}$ denotes the tensor of the
energy-momentum of matter, specifically

$\mathcal{T}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}}\;\;\;\;.$ (11)

In the given expression, we have $f$ defined as $f(\mathbb{Q})$, and
$f_{\mathbb{Q}}$ as its first derivative with respect to $\mathbb{Q}$. It is
worth noting that the matter Lagrangian density is varied independently
regarding the connection, resulting in the absence of hyper-momentum.
Furthermore, as is widely recognized, the outcomes of GR (in the framework of
Scalar-Tensor Extended General Relativity, STEGR) are obtained by setting
$f(\mathbb{Q})=\mathbb{Q}$. Consequently, the Lagrangian density takes the
form $\mathcal{L}=-\frac{\mathbb{Q}}{2\kappa^{2}}+\mathcal{L}_{m}$.

##  III Static AdS/dS black hole solution

We utilize the equations of motions of $f(\mathbb{Q})$ gravity, as expressed
by (II), to investigate the cylindrical $N$-dimensional spacetime. This
analysis yields the following line element, presented in cylindrical
coordinates ($t$, $r$, $\xi_{1}$, $\xi_{2}$,$\cdots$ $\xi_{N-2}$), as
described in Awad et al. (2017):

$\displaystyle
ds{}^{2}=h(r)dt^{2}-\frac{dr^{2}}{h_{1}(r)}-r^{2}\sum_{i=1}^{N-2}d\xi^{2}_{i}\,.$
(12)

Here, $h(r)$ and $h_{1}(r)$ represent two unknowns dependent on $r$.
Furthermore, the equation for $\mathbb{Q}$ concerning the spacetime, as
provided by Eq. (12), is computed in N-dimensions and yields

$\displaystyle\mathbb{Q}=-\frac{(N-2)h_{1}[(N-3)h+h^{\prime}]}{r^{2}h}\,.$
(13)

Utilizing Eq. (12) to the equations of motions (II) when
${{{\mathcal{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}=0$
we obtain the ensuing non-zero components:

$\displaystyle\zeta_{t}{}^{t}\equiv\frac{1}{2r^{2}h^{2}}\left[2(N-2)^{2}h_{1}{}^{2}r^{2}f_{\mathbb{Q}\mathbb{Q}}[h^{\prime
2}-hh^{\prime\prime}]+(N-2)rhh_{1}h^{\prime}[2(N-2)f_{\mathbb{Q}\mathbb{Q}}\\{h_{1}-rh^{\prime}_{1}\\}+r^{2}f_{\mathbb{Q}}]+h\left\\{(N-2)rhh^{\prime}_{1}\left[r^{2}f_{\mathbb{Q}}\right.\right.\right.$
$\displaystyle\left.\left.\left.-2(N-3)(N-2)f_{\mathbb{Q}\mathbb{Q}}h_{1}\right]+h(r^{4}f(\mathbb{Q})+2(N-2)(N-3)r^{2}h_{1}f_{\mathbb{Q}}+4(N-3)(N-2)^{2}h_{1}{}^{2}f_{\mathbb{Q}\mathbb{Q}})\right\\}\right]=0\,,$
$\displaystyle\zeta_{r}{}^{r}\equiv\frac{f(\mathbb{Q})r^{2}h+2(N-2)rf_{\mathbb{Q}}h_{1}h^{\prime}+2(N-2)(N-3)f_{\mathbb{Q}}hh_{1}}{2r^{2}h}=0\,,$
$\displaystyle\zeta_{\xi_{1}}{}^{\xi_{1}}=\zeta_{\xi_{2}}{}^{\xi_{2}}=\cdots\cdots=\zeta_{\xi_{N-2}}{}^{\xi_{N-2}}\equiv\frac{1}{4r^{4}h^{3}}\left\\{2r^{2}h_{1}hh^{\prime\prime}[h(r^{2}f_{\mathbb{Q}}-2(N-2)(N-3)f_{\mathbb{Q}\mathbb{Q}}h_{1})-(N-2)rh_{1}h^{\prime}f_{\mathbb{Q}\mathbb{Q}}]\right.$
$\displaystyle\left.+2(N-2)f_{\mathbb{Q}\mathbb{Q}}r^{3}h_{1}{}^{2}h^{\prime
3}-r^{2}hh_{1}h^{\prime
2}[r^{2}f_{\mathbb{Q}}+2f_{\mathbb{Q}\mathbb{Q}}(N-2)(rh^{\prime}_{1}-(2N-5)h_{1})]+rh^{2}h^{\prime}\left[rh^{\prime}_{1}(r^{2}f_{\mathbb{Q}}-6(N-2)(N-3)f_{\mathbb{Q}\mathbb{Q}}h_{1})\right.\right.$
$\displaystyle\left.\left.+2(2N-5)r^{2}f_{\mathbb{Q}}h_{1}+8(N-2)(N-3)f_{\mathbb{Q}\mathbb{Q}}h_{1}{}^{2}\right]+2h^{2}\left[(N-3)rhh^{\prime}_{1}(r^{2}f_{\mathbb{Q}}-2(N-2)(N-3)h_{1}f_{\mathbb{Q}\mathbb{Q}})+h\left(r^{4}f(\mathbb{Q})\right.\right.\right.$
$\displaystyle\left.\left.\left.+2(N-3)^{2}r^{2}h_{1}f_{\mathbb{Q}}+4(N-2)(N-3)^{2}f_{\mathbb{Q}\mathbb{Q}}h_{1}{}^{2}\right)\right]\right\\}=0\,.$
(14)

and the second equation of Eq. (II) takes the form:

$\displaystyle{\frac{2\,rh_{1}q^{\prime\prime}h+\left(h\left(h^{\prime}_{1}r+2(N-2)\,h_{1}\right)-h^{\prime}h_{1}r\right)q^{\prime}}{2h^{2}r}}=0\,.$
(15)

Next, we will determine a comprehensive solution to Eqs. (IV) by employing a
particular shape of $f(\mathbb{Q})$, i.e.,

$\displaystyle
f(\mathbb{Q})=\mathbb{Q}+\frac{1}{2}\alpha\mathbb{Q}^{2}-2\Lambda\,,$ (16)

where $\alpha$ is a dimensional constant that has the unite of
${\textit{l}ength^{2}}$ and $\Lambda$ is the cosmological constant. In the
context of this particular configuration of $f(\mathbb{Q})$, Eqs. (13) yeilds:

$\displaystyle\zeta_{t}{}^{t}\equiv\frac{1}{2r^{2}h^{2}}\left[\alpha
2(N-2)^{2}h_{1}{}^{2}r^{2}[h^{\prime
2}-hh^{\prime\prime}]+\alpha(N-2)rhh_{1}h^{\prime}[2(N-2)\\{h_{1}-rh^{\prime}_{1}\\}+r^{2}\\{1+\alpha\mathbb{Q}\\}]+h\left\\{(N-2)rhh^{\prime}_{1}\left[r^{2}\right.\right.\right.$
$\displaystyle\left.\left.\left.\times\\{1+\alpha\mathbb{Q}\\}-2\alpha(N-3)(N-2)h_{1}\right]+h\right(r^{4}\\{\mathbb{Q}+1/2\alpha\mathbb{Q}^{2}-2\Lambda\\}+2(N-2)(N-3)r^{2}h_{1}\\{1+\alpha\mathbb{Q}\\}\right.$
$\displaystyle\left.\left.\left.+4\alpha(N-3)(N-2)^{2}h_{1}{}^{2}\right)\right\\}\right]=0\,,$
$\displaystyle\zeta_{r}{}^{r}\equiv\frac{r^{2}\\{1+\alpha\mathbb{Q}\\}h+2(N-2)rh_{1}h^{\prime}\\{1+\alpha\mathbb{Q}\\}+2(N-2)(N-3)hh_{1}\\{1+\alpha\mathbb{Q}\\}}{2r^{2}h}=0\,,$
$\displaystyle\zeta_{\xi_{1}}{}^{\xi_{1}}\equiv\zeta_{\xi_{2}}{}^{\xi_{2}}=\cdots\cdots=\zeta_{\xi_{N-2}}{}^{\xi_{N-2}}=\frac{1}{4r^{4}h^{3}}\left\\{2r^{2}h_{1}hh^{\prime\prime}[h(r^{2}\\{1+\alpha\mathbb{Q}\\}-2\alpha(N-2)(N-3)h_{1})-(N-2)\alpha
rh_{1}h^{\prime}]\right.$
$\displaystyle\left.+2\alpha(N-2)r^{3}h_{1}{}^{2}h^{\prime
3}-r^{2}hh_{1}h^{\prime
2}[r^{2}\\{1+\alpha\mathbb{Q}\\}+2\alpha(N-2)(rh^{\prime}_{1}-(2N-5)h_{1})]+rh^{2}h^{\prime}\left[rh^{\prime}_{1}(r^{2}\\{1+\alpha\mathbb{Q}\\}\right.\right.$
$\displaystyle\left.\left.-6\alpha(N-2)(N-3)h_{1})+2(2N-5)r^{2}h_{1}\\{1+\alpha\mathbb{Q}\\}+8\alpha(N-2)(N-3)h_{1}{}^{2}\right]+2h^{2}\left[(N-3)rhh^{\prime}_{1}\left(r^{2}\\{1+\alpha\mathbb{Q}\\}\right.\right.\right.$
$\displaystyle\left.\left.\left.-2\alpha(N-2)(N-3)h_{1}\right)+h\left(r^{4}\\{\mathbb{Q}+1/2\alpha\mathbb{Q}^{2}-2\Lambda\\}+2(N-3)^{2}r^{2}h_{1}\\{1+\alpha\mathbb{Q}\\}+4\alpha(N-2)(N-3)^{2}h_{1}{}^{2}\right)\right]\right\\}=0\,.$
(17)

A solution in the general $N$-dimensional case for Eq. (III) is:

$\displaystyle
h(r)=-\frac{r^{2}(1\pm\sqrt{1-12\alpha\Lambda})}{3(N-1)(N-2)\alpha}+\frac{c_{1}}{r^{D-3}},\qquad\quad
h_{1}(r)=h(r)\,.$

In this context, $c_{1}$ represents a dimensional constant of integration. For
the sake of simplifying the calculations, we will make the assumption that

$\displaystyle\Lambda=\frac{1}{12\alpha}\,.$ (19)

This assumption leads to a unique solution that can be expressed in the
form333These solutions exhibit two possible values for the cosmological
constant: $\frac{1\pm\sqrt{1-12\alpha\Lambda}}{12\alpha}$.

$\displaystyle h(r)=-\frac{r^{2}}{3(N-1)(N-2)\alpha}+\frac{c_{1}}{r^{N-3}}\,.$
(20)

Equation (III) clearly illustrates that the higher order of $f(\mathbb{Q})$ in
the case of quadratic form acts as a cosmological constant.

##  IV Charged AdS/dS black hole solution

Utilizing Eq. (12) to the equations of motions (II) as
${{{\mathcal{T}}^{{}^{{}^{{\\!\\!\\!\\!\scriptstyle{em}}}}}}}^{\nu}_{\mu}\neq
0$ we obtain the subsequent non-zero components as:

$\displaystyle\zeta_{t}{}^{t}\equiv\frac{1}{2r^{2}h^{2}}\left[2(N-2)^{2}h_{1}{}^{2}r^{2}f_{\mathbb{Q}\mathbb{Q}}[h^{\prime
2}-hh^{\prime\prime}]+(N-2)rhh_{1}h^{\prime}[2(N-2)f_{\mathbb{Q}\mathbb{Q}}\\{h_{1}-rh^{\prime}_{1}\\}+r^{2}f_{\mathbb{Q}}]+h\left\\{(N-2)rhh^{\prime}_{1}\left[r^{2}f_{\mathbb{Q}}\right.\right.\right.$
$\displaystyle\left.\left.\left.-2(N-3)(N-2)f_{\mathbb{Q}\mathbb{Q}}h_{1}\right]+h(r^{4}f(\mathbb{Q})+2(N-2)(N-3)r^{2}h_{1}f_{\mathbb{Q}}+4(N-3)(N-2)^{2}h_{1}{}^{2}f_{QQ})\right\\}-2q^{\prime
2}r^{4}hh_{1}\right]=0\,,$
$\displaystyle\zeta_{r}{}^{r}\equiv\frac{f(\mathbb{Q})r^{2}h+2(N-2)rf_{\mathbb{Q}}h_{1}h^{\prime}+2(N-2)(N-3)f_{\mathbb{Q}}hh_{1}-2q^{\prime
2}r^{4}hh_{1}}{2r^{2}h}=0\,,$
$\displaystyle\zeta_{\phi_{1}}{}^{\phi_{1}}=\zeta_{\phi_{2}}{}^{\phi_{2}}=\cdots\cdots=\zeta_{\phi_{N-2}}{}^{\phi_{N-2}}\equiv\frac{1}{4r^{4}h^{3}}\left\\{2r^{2}h_{1}hh^{\prime\prime}[h(r^{2}f_{\mathbb{Q}}-2(N-2)(N-3)f_{\mathbb{Q}\mathbb{Q}}h_{1})-(N-2)rh_{1}h^{\prime}f_{\mathbb{Q}\mathbb{Q}}]\right.$
$\displaystyle\left.+2(N-2)f_{\mathbb{Q}\mathbb{Q}}r^{3}h_{1}{}^{2}h^{\prime
3}-r^{2}hh_{1}h^{\prime
2}[r^{2}f_{\mathbb{Q}}+2f_{\mathbb{Q}\mathbb{Q}}(N-2)(rh^{\prime}_{1}-(2N-5)h_{1})]+rh^{2}h^{\prime}\left[rh^{\prime}_{1}(r^{2}f_{\mathbb{Q}}-6(N-2)(N-3)f_{\mathbb{Q}\mathbb{Q}}h_{1})\right.\right.$
$\displaystyle\left.\left.+2(2N-5)r^{2}f_{\mathbb{Q}}h_{1}+8(N-2)(N-3)f_{\mathbb{Q}\mathbb{Q}}h_{1}{}^{2}\right]+2h^{2}\left[(N-3)rhh^{\prime}_{1}(r^{2}f_{\mathbb{Q}}-2(N-2)(N-3)h_{1}f_{\mathbb{Q}\mathbb{Q}})+h\left(r^{4}f(\mathbb{Q})\right.\right.\right.$
$\displaystyle\left.\left.\left.+2(N-3)^{2}r^{2}h_{1}f_{\mathbb{Q}}+4(N-2)(N-3)^{2}f_{\mathbb{Q}\mathbb{Q}}h_{1}{}^{2}+4q^{\prime
2}r^{4}h^{2}h_{1}\right)\right]\right\\}=0\,.$ (21)

Finally, Eq. (II) yields:

$\displaystyle\frac{rq^{\prime}[h_{1}h^{\prime}-hh^{\prime}_{1}]-2rhh_{1}q^{\prime\prime}-2(N-2)hh_{1}q^{\prime}}{2rh^{2}}=0\,.$
(22)

We will now seek a comprehensive solution to the aforementioned differential
equations.? By employing a particular format for $f(\mathbb{Q})$ defined by
Eq. (16) we get:

$\displaystyle\zeta_{t}{}^{t}=\frac{1}{2r^{2}h^{2}}\left[\alpha
2(N-2)^{2}h_{1}{}^{2}r^{2}[h^{\prime
2}-hh^{\prime\prime}]+\alpha(N-2)rhh_{1}h^{\prime}[2(N-2)\\{h_{1}-rh^{\prime}_{1}\\}+r^{2}\\{1+\alpha\mathbb{Q}\\}]+h\left\\{(N-2)rhh^{\prime}_{1}\left[r^{2}\right.\right.\right.$
$\displaystyle\left.\left.\left.\times\\{1+\alpha\mathbb{Q}\\}-2\alpha(N-3)(N-2)h_{1}\right]+h\right(r^{4}\\{\mathbb{Q}+1/2\alpha\mathbb{Q}^{2}-2\Lambda\\}+2(N-2)(N-3)r^{2}h_{1}\\{1+\alpha\mathbb{Q}\\}\right.$
$\displaystyle\left.\left.\left.+4\alpha(N-3)(N-2)^{2}h_{1}{}^{2}\right)\right\\}-2q^{\prime
2}r^{4}hh_{1}\right]\,,$
$\displaystyle\zeta_{r}{}^{r}=\frac{r^{2}\\{1+\alpha\mathbb{Q}\\}h+2(N-2)rh_{1}h^{\prime}\\{1+\alpha\mathbb{Q}\\}+2(N-2)(N-3)hh_{1}\\{1+\alpha\mathbb{Q}\\}-2q^{\prime
2}r^{4}hh_{1}}{2r^{2}h}\,,$
$\displaystyle\zeta_{\phi_{1}}{}^{\phi_{1}}=\zeta_{\phi_{2}}{}^{\phi_{2}}=\cdots\cdots=\zeta_{\phi_{N-2}}{}^{\phi_{N-2}}=\frac{1}{4r^{4}h^{3}}\left\\{2r^{2}h_{1}hh^{\prime\prime}[h(r^{2}\\{1+\alpha\mathbb{Q}\\}-2\alpha(N-2)(N-3)h_{1})-(N-2)\alpha
rh_{1}h^{\prime}]\right.$
$\displaystyle\left.+2\alpha(N-2)r^{3}h_{1}{}^{2}h^{\prime
3}-r^{2}hh_{1}h^{\prime
2}[r^{2}\\{1+\alpha\mathbb{Q}\\}+2\alpha(N-2)(rh^{\prime}_{1}-(2N-5)h_{1})]+rh^{2}h^{\prime}\left[rh^{\prime}_{1}(r^{2}\\{1+\alpha\mathbb{Q}\\}\right.\right.$
$\displaystyle\left.\left.-6\alpha(N-2)(N-3)h_{1})+2(2N-5)r^{2}h_{1}\\{1+\alpha\mathbb{Q}\\}+8\alpha(N-2)(N-3)h_{1}{}^{2}\right]+2h^{2}\left[(N-3)rhh^{\prime}_{1}\left(r^{2}\\{1+\alpha\mathbb{Q}\\}\right.\right.\right.$
$\displaystyle\left.\left.\left.-2\alpha(N-2)(N-3)h_{1}\right)+h\left(r^{4}\\{\mathbb{Q}+1/2\alpha\mathbb{Q}^{2}-2\Lambda\\}+2(N-3)^{2}r^{2}h_{1}\\{1+\alpha\mathbb{Q}\\}+4\alpha(N-2)(N-3)^{2}h_{1}{}^{2}\right)\right]+4q^{\prime
2}r^{4}h^{2}h_{1}\right\\}\,.$ (23)

A general $N$-dimension solution of Eqs. (22) and (IV) has the form:

$\displaystyle
h(r)=\frac{r^{2}}{3(N-1)(N-2)\left|\alpha\right|}-\frac{M}{r^{N-3}}+\frac{3(N-3)v^{2}}{(N-2)r^{2(N-3)}}+\frac{2\sqrt{3\left|\alpha\right|}(N-3)^{3}v^{3}}{(2N-5)(N-2)r^{3N-8}},$
$\displaystyle h_{1}(r)=h(r)h_{2}(r),\qquad
h_{2}(r)=\frac{1}{\left[1+\frac{v\sqrt{3\left|\alpha\right|}}{(N-3)r^{N-2}}\right]^{2}},\qquad
q(r)=\frac{v}{r^{N-3}}+\frac{(N-3)^{2}v{}^{2}\sqrt{3\left|\alpha\right|}}{(2N-5)r^{2N-5}}\,.$

Here, we assign the values of $c_{1}$ and $c_{2}$ as $M=-c_{1}$, $v=c_{2}$,
which signifies the momentum associated with the monopole. The expression for
the quadrupole momentum is given by:

$c_{3}=\frac{(N-3)^{2}v^{2}\sqrt{3\left|\alpha\right|}}{(2N-5)}.$

It is worth noting that Eq. (IV) indicates that $\alpha$ must have a value in
the negative range; otherwise, an impractical solution is derived.

As evident from Eq. (IV), the electric potential $q(r)$ is contingent on both
monopole and quadrupole moments. When setting $v=0$, both moments disappear,
resulting in a non-charged solution. This implies that the quadratic form of
$f(\mathbb{Q})$ does not allow for a Reissner-Nordström solution within the
linear framework of the non-metricity theory.

##  V The key characteristics of the black hole solutions (IV)

Now, let’s examine some pertinent aspects of the solution with charge outlined
in the preceding section. The line-element of the solution (IV) is expressed
as:

$\displaystyle
ds{}^{2}=\Biggl{[}r^{2}\Lambda_{eff}-\frac{M}{r^{N-3}}+\frac{3(N-3)v^{2}}{(N-2)r^{2(N-3)}}+\frac{2\sqrt{3\left|\alpha\right|}(N-3)^{3}v^{3}}{(2N-5)(N-2)r^{3N-8}}\Biggr{]}dt^{2}$
$\displaystyle-$
$\displaystyle\frac{dr^{2}}{\left[1+\frac{v\sqrt{3\left|\alpha\right|}}{(N-3)r^{N-2}}\right]^{2}\Biggr{[}r^{2}\Lambda_{eff}-\frac{M}{r^{N-3}}+\frac{3(N-3)v^{2}}{(N-2)r^{2(N-3)}}+\frac{2\sqrt{3\left|\alpha\right|}(N-3)^{3}v^{3}}{(2N-5)(N-2)r^{3N-8}}\Biggr{]}}-r^{2}\sum_{i=1}^{N-2}d\phi^{2}_{i}\,.$

Here $\Lambda_{eff}=\frac{1}{3(N-1)(N-2)\left|\alpha\right|}$. Equation (IV)
distinctly indicates that the line-element of the solution with charge
asymptotes as AdS spacetime. It’s worth noting that there is no equivalent for
the linear form of the non-metricity solution when approaching the limit
$\alpha\rightarrow 0$. This implies that the charged solution lacks an
analogue in the lower order of the non-metricity theory. In the limit as
$v\rightarrow 0$, we recover the non-charged black holes discussed with
asymptotically AdS behavior discussed above. It’s worth noting that despite
having different components for the metric, specifically $g_{tt}$ and
$g^{rr}$, these asymptotically AdS charged solutions share common event and
Killing horizons.

Singularity:
In this context, we recognize tangible singularities by evaluating all
potential invariants in the context of the $f(\mathbb{Q})$ theory. Considering
the function $h(r)$ may have roots, denoted as $r_{n}$, it becomes imperative
to examine the behavior of invariants in the vicinity of these roots. Upon
computing the various invariants, we arrive at:

$\displaystyle\mathbb{Q}=\frac{1}{3\left|\alpha\right|}+\frac{2(N-3)v}{3\left|\alpha\right|r^{N-2}}\,,$
$\displaystyle\mathbb{R}^{\mu\nu\lambda\rho}\mathbb{R}_{\mu\nu\lambda\rho}\approx\frac{2N}{9(N-1)(N-2)^{2}\alpha^{2}}-\frac{16(N-3)v}{9(N-2)^{2}(N-1)\sqrt{\left|\alpha\right|^{3}}r^{N-2}}+{\mathcal{O}}\left(\frac{1}{r^{2(N-2)}}\right)\,,$
$\displaystyle\mathbb{R}^{\mu\nu}\mathbb{R}_{\mu\nu}\approx\frac{N}{9(N-2)^{2}\alpha^{2}}-\frac{4(N-3)v}{3(N-2)^{2}\sqrt{\left|\alpha\right|^{3}}r^{N-2}}+{\mathcal{O}}\left(\frac{1}{r^{2(N-2)}}\right)\,,$
$\displaystyle\mathbb{R}\approx-\frac{N}{3(N-2)\alpha}+\frac{4(N-3)v}{3(N-2)\sqrt{\left|\alpha\right|^{3}}r^{N-2}}+{\mathcal{O}}\left(\frac{1}{r^{2N-3}}\right)\,,$
$\displaystyle\mathbb{Q}^{\mu\nu\rho}\mathbb{Q}_{\mu\nu\rho}\approx\frac{4N}{3(N-1)(N-2)\alpha}-\frac{16(N-3)v}{3(N-2)(N-1)\sqrt{\left|\alpha\right|}r^{N-2}}+{\mathcal{O}}\left(\frac{1}{r^{N-1}}\right)\,,$
$\displaystyle{\mathbb{Q}}_{\mu}{\mathbb{Q}}^{\mu}\approx\frac{4(N-2)}{3(N-1)\left|\alpha\right|}-\frac{16(N-2)(N-3)v}{3(N-1)\sqrt{\left|\alpha\right|}r^{(N-2)}}+{\mathcal{O}}\left(\frac{1}{r^{(N-1)}}\right)\,,$
$\displaystyle\tilde{\mathbb{Q}}_{\mu}\tilde{\mathbb{Q}}^{\mu}\approx\frac{4(N-2)(N-3)^{2}v^{2}}{27(N-1)\alpha^{2}r^{2(N-2)}}+{\mathcal{O}}\left(\frac{1}{r^{3(N-2)}}\right)\,,$
(26)

where $\mathbb{R}^{\mu\nu\lambda\rho}\mathbb{R}_{\mu\nu\lambda\rho}$,
$\mathbb{R}^{\mu\nu}\mathbb{\mathbb{Q}}_{\mu\nu}$, $\mathbb{Q}$,
$\mathbb{Q}^{\mu\nu\lambda}\mathbb{Q}_{\mu\nu\lambda}$
$\mathbb{Q}^{\mu}\mathbb{Q}_{\mu}$,
$\tilde{\mathbb{Q}}^{\mu}\tilde{\mathbb{Q}}_{\mu}$ and $\mathbb{Q}$ are all
the possible invariants that can be constructed in this theory444 Here
Kretschmann scalar, the Ricci tensor square, the Ricci scalar, the non-
metricity tensor square, the non-metricity square vectors and the non-
metricity scalar, are defined as:
$\mathbb{R}^{\mu\nu\lambda\rho}\mathbb{R}_{\mu\nu\lambda\rho}$,
$\mathbb{R}^{\mu\nu}\mathbb{\mathbb{Q}}_{\mu\nu}$, $\mathbb{Q}$,
$\mathbb{Q}^{\mu\nu\lambda}\mathbb{Q}_{\mu\nu\lambda}$
$\mathbb{Q}^{\mu}\mathbb{Q}_{\mu}$,
$\tilde{\mathbb{Q}}^{\mu}\tilde{\mathbb{Q}}_{\mu}$ and $\mathbb{Q}$,
respectively The aforementioned invariants reveal that:
a) A singularity is present at $r=0$, characterized as a singularity in
curvature.
b) For the solution with charge, the singularity is indicated by the non-
metricity scalar at $r=0$. In the vicinity of $r=0$, and in the charged case,
the behavior of
$K=\mathbb{R}_{\mu\nu}\mathbb{R}^{\mu\nu}=\mathbb{R}=\mathbb{Q}^{\alpha\beta\gamma}\mathbb{Q}_{\alpha\beta\gamma}=\mathbb{Q}^{\alpha}\mathbb{Q}_{\alpha}=\tilde{\mathbb{Q}}^{\alpha}\tilde{\mathbb{Q}}_{\alpha}=\mathbb{Q}\sim
r^{-(N-2)}$, unlike the solutions of the of the non-metricity-Maxwell theory
which have $K=r^{-3(N-2)}$, and
$\mathbb{Q}^{\alpha\beta\gamma}\mathbb{Q}_{\alpha\beta\gamma}=\mathbb{Q}^{\alpha}\mathbb{Q}_{\alpha}\approx
r^{-(N-1)}$. This unequivocally demonstrates that the singularity is
considerably less severe compared to the one obtained in the linear form of
non-metricity for the solution with charge. Such outcome prompts the following
question: If these singularities align with the concept of “weak
singularities” as defined by Tipler and Krolak Tipler (1977); Clarke and
Królak (1985).? Additionally, it prompts inquiry into the potential to extend
geodesics beyond these areas, a topic slated for discussion in future studies.

##  VI The thermodynamic characteristics of the black hole described by Eq.
(IV)

To investigate the thermodynamic characteristics of the newly discovered
solution specified in Eq. (IV), we introduced the notion of the Hawking
temperature Nashed and Nojiri (2023); Mazharimousavi (2023)555Due to the
inequality of the metric potentials in the solution given by Eq. (IV) the
Hawking temperature deviates from the conventional one where the metric
potentials are equal Sheykhi (2012, 2010); Hendi et al. (2010); Nashed and
Nojiri (2023); Sheykhi et al. (2010) as:

$T_{1}=\frac{h^{\prime}(r_{1})}{4\pi\sqrt{-h(r_{1})h_{1}(r_{1})}}\,.$ (27)

In this context, the symbol ′ indicates a derivative with respect to the event
horizon, situated at the value of r equal to $r_{1}$. This value of $r_{1}$
represents the most significant positive root of $h_{1}(r_{1})=0$, while
ensuring that $h^{\prime}_{1}(r_{1})$ is not equal to zero. The Bekenstein-
Hawking entropy of $f(\mathbb{Q})$ theory is represented as Cognola et al.
(2011); Zheng and Yang (2018)666Keep in mind that the entropy framed in linear
non-metricity theory differs from that framed in $f(\mathbb{Q})$ geocentric
theory. When $f(\mathbb{Q})=\mathbb{Q}$ we get the will know of the non-
metricity theory.:

$S(r_{1})=\frac{1}{4}A{f_{\mathbb{Q}}}(r_{1})\,.$ (28)

In this frame, $A$ represents the surface area of the event horizon. The black
hole will be thermodynamically stable according to the heat capacity
indicator, which is denoted as $C_{1}$; if $C_{1}>0$, then it is stable, and
if $C_{1}<0$, then it will not be stable. In the subsequent analysis, we
evaluate the thermal stability of these black hole solutions by observing the
behavior of their individual heat capacities Nouicer (2007); Chamblin et al.
(1999)

$C_{1}=\frac{dE_{1}}{dT_{1}}=\frac{\partial M}{\partial
r_{1}}\left(\frac{\partial T}{\partial r_{1}}\right)^{-1}\,.$ (29)

In this context, $E_{1}$ represents the energy. Ultimately, the Gibbs free
energy is established as followsZheng and Yang (2018); Kim and Kim (2012):

$G_{1}=E_{1}-T_{1}S_{+}\,.$ (30)

In the context of the solution provided in Eq. (IV) and within the framework
of four dimensions, the horizons are obtained as:

$\displaystyle
r=\sqrt[6]{6z^{6}\Lambda_{eff}+2v^{3}\sqrt{\left|\alpha\right|}+9v^{2}z^{2}-6Mr^{3}}.$
(31)

Additionally, from Eq. (IV), we can derive the following expression for mass:

$\displaystyle
M=\sqrt[6]{6z^{6}\Lambda_{eff}+2v^{3}\sqrt{\left|\alpha\right|}+9v^{2}z^{2}-6Mr^{3}}.$
(32)

Equation (34) illustrates that the horizon affects the black hole’s total
mass, $v$, and the effective cosmological constant $\Lambda_{eff}$. The
relationship between $h_{1}(r)$ and $r$ is depicted in Fig.1 0(a),
demonstrating the possible horizons of such solution for $v=0$ and $v\neq 0$.
Moreover, Fig. 1 0(b) shows the distinct areas of the horizons, where the blue
curve represents the two horizons, the degenerate horizon is determined by
equating $\frac{\partial M_{h}}{\partial r_{h}}$ to zero (depicted by the red
curve), and the green curve represents the naked singularity zone when
$M=0.3$.

(a) The characteristics of the metric potential $g_{rr}$

(b) The locations of the horizons in the metric potential $g_{rr}$

(c) The characteristics of the entropy

(d) The temperature according to Hawking

(e) The characteristics of the Gibbs function

Figure 1: 0(a) The overall trends of the metric potential $g_{rr}$ are
illustrated in Fig. 1 for both $v=0$ and $v\neq 0$; 0(b) presents the general
tendencies of $g_{rr}$ as $r\to\infty$, revealing three distinct regions; 0(c)
showcases the behavior of the entropy; 0(d) depicts the variations in the
Hawking temperature; and 0(e) illustrates the pattern of the Gibbs function.
All the parameters characterized the model take the values
$\Lambda_{eff}=0.1$, $M=v=1$ and $\alpha=-0.1$.

Utilizing Eq. (28), we can determine the entropy associated with the described
black hole solution of Eq. (IV) as follows:

$\displaystyle{S_{1}}=\frac{2\pi\left(v\,{r_{1}}^{2}+\sqrt{\left|\alpha\right|}\right)}{3v}.$
(33)

Examining Eq. (33), it is evident that the charge $v$ is not permitted to
assume a zero value. When $\alpha=0$, we cannot recover the typical entropy of
GR, primarily because of the constraint imposed by Eq. (19). The entropy
trends are illustrated in Fig.1 0(c), showcasing a well-behaved pattern of the
entropy.

The Hawking temperature linked to the black hole solution of Eq. (IV) is
calculated as:

$\displaystyle{T_{1}}=\frac{\left(6\,{r_{1}}^{6}\Lambda_{eff}-4\,{v}^{3}\sqrt{3}\sqrt{\left|\alpha\right|}-9\,{v}^{2}{r_{1}}^{2}+3\,M{r_{1}}^{3}\right)}{2\pi{r_{1}}^{5}\sqrt{\left({r_{1}}^{2}\Lambda_{eff}+{\frac{{v}^{3}\sqrt{3}\sqrt{\alpha}}{3{r_{1}}^{4}}}+{\frac{3{v}^{2}}{2{r_{1}}^{2}}}-{\frac{M}{r_{1}}}\right)^{2}\left(1+{\frac{v\sqrt{3}\sqrt{\alpha}}{{r_{1}}^{2}}}\right)^{2}}}\,.$
(34)

Here ${T_{1}}$ represent the temperature of Hawking at the event horizon. The
temperature is depicted in Fig. 0(d), revealing that it is always positive.

The grand canonical ensemble’s free energy, which is known as Gibbs free
energy, is described as Zheng and Yang (2018); Kim and Kim (2012):

$G(r_{+})=M(r_{1})-T(r_{1})S(r_{1}),$ (35)

where $T(r_{h})$, $S(r_{h})$, and $E(r_{h})$ represent the temperature,
entropy, and quasilocal energy at the event horizon, respectively. By
substituting Eqs. (28), (31), (33), and (34) into (35), we obtain:

$\displaystyle
G(r_{+})=\frac{1}{6}\left\\{36v{r_{1}}^{14}{\Lambda_{eff}}^{2}+36{r_{1}}^{12}{\Lambda_{eff}}^{2}{v}^{2}\sqrt{3}\sqrt{\alpha}+132{r_{1}}^{8}\Lambda_{eff}{v}^{4}\sqrt{3}\sqrt{\alpha}+72{r}^{6}\Lambda_{eff}{v}^{5}\alpha+108{r_{1}}^{10}\Lambda_{eff}{v}^{3}-36v{r}^{11}\Lambda_{eff}M\right.$
$\displaystyle\left.-36{r_{1}}^{9}\Lambda_{eff}M{v}^{2}\sqrt{3}\sqrt{\alpha}+120{v}^{7}\alpha{r_{1}}^{2}+12{v}^{8}\sqrt{3}{\alpha}^{3/2}+117{v}^{6}\sqrt{3}\sqrt{\alpha}{r_{1}}^{4}-66{v}^{4}\sqrt{3}\sqrt{\alpha}M{r_{1}}^{5}-36{v}^{5}\alpha
M{r_{1}}^{3}+81{v}^{5}{r_{1}}^{6}\right.$
$\displaystyle\left.-54{v}^{3}{r_{1}}^{7}M-12{r_{1}}^{12}\Lambda_{eff}v-12{r_{1}}^{10}\Lambda_{eff}\sqrt{3}\sqrt{\alpha}+8{r_{1}}^{6}{v}^{4}\sqrt{3}\sqrt{\alpha}+24{r_{1}}^{4}{v}^{3}\alpha+18{r_{1}}^{8}{v}^{3}+18{r_{1}}^{6}{v}^{2}\sqrt{3}\sqrt{\alpha}-6{r_{1}}^{9}Mv\right.$
$\displaystyle\left.-6{r}^{7}M\sqrt{3}\sqrt{\alpha}\right\\}\left\\{{r_{1}}^{3}\left(6{r_{1}}^{6}\Lambda_{eff}+2{v}^{3}\sqrt{3}\sqrt{\alpha}+9{v}^{2}{r}^{2}-6M{r}^{3}\right)\left({r_{1}}^{2}+v\sqrt{3}\sqrt{\alpha}\right)v\right\\}^{-1}\,.$
(36)

The trends of the Gibbs energy for the black hole (IV) are depicted in Fig.
10(e) for specific values of the model parameters. Fig. 10(e) illustrates that
the Gibbs free energy is consistently positive, indicating greater global
stability.

##  VII Solutions with multiple horizons

The most basic black hole solution is characterized by the Ads/dS-
Schwarzschild metric, where the description of the metric coefficient $g_{00}$
is:

$f=\Lambda r^{2}-\frac{2M}{r}=f_{1}(r)(r-r_{1})(r-r_{2})(r-r_{3})\,.$ (37)

Here, $f_{1}(r)=\frac{1}{r}$ and $r_{i},i=1,2,3$ represent the horizons.
Equation (37) possesses a single real horizon at
$r=r_{1}=\sqrt[3]{2M\Lambda}$, while the other two horizons, $r_{2}$ and
$r_{3}$, are imaginary. Equation (37) can be derived from Eq. (IV) when $v=0$.
When $v\neq 0$, we derive the solution of Eq. (IV) in the case of four-
dimensions in the following manner:

$h=r^{2}\Lambda_{eff}-\frac{M}{r}+\frac{3v^{2}}{2r^{2}}+\frac{\sqrt{3\left|\alpha\right|}v^{3}}{3r^{4}}=h_{2}(r)(r-r_{1})(r-r_{2})(r-r_{3})(r-r_{4})(r-r_{5})(r-r_{6})\,.$
(38)

We identify the impact of the quadratic form of non-metricity, generating six
roots from which at least three can be extracted. Here,
$h_{2}(r)=\frac{1}{r^{4}}$. Now we are going to show that, it is possible to
produce at least three real roots of Eq. (38) in the following manner:

$h=r^{2}\Lambda_{eff}-\frac{M}{r}+\frac{3v^{2}}{2r^{2}}+\frac{\sqrt{3\left|\alpha\right|}v^{3}}{3r^{4}}=h_{2}(r)(r-r_{1})(r-r_{2})(r-r_{3})\,.$
(39)

In this study, $r_{1}$ and $r_{2}$ stand for the cosmological and event
horizons, respectively, while $r_{3}$ is the radius horizon obtained from the
contribution of the quadratic form of non-metricity. Deriving the explicit
forms of the three horizons for Eq. (IV) poses a challenge. Hence, we will
numerically and graphically solve Eq. (IV) in cases where $v<0$. We graph Eq.
(IV) regarding certain values of mass, charge, and parameter $\alpha$ related
to the quadratic form of non-metricity.

(a) The BH’s multi-horizon Figure 2:  Multi-horizon plot of Eq. (IV) against
coordinate $r$.

In the case of a black hole with one or two horizons, the curvature scalar is
recognized to be zero, as observed in the Schwarzschild and Reissner-
Nordstr”om solutions. However, for a black hole with more than two horizons,
the curvature scalar becomes singular at the center. At $r=0$, the Kretschmann
scalar is singular for any black hole with more than one horizon. This is
explained in detail by Eq. (V)..

Let’s now examine the thermodynamics of the multi-horizons stated earlier.
Because of the negative value of $v$ in the case of multi-horizons, we obtain
for BH (IV) that these quantities behave differently thermodynamically. We
have stable BHs for a multi-horizon spacetime, as demonstrated by all the
charts in Fig. 3. The Hawking temperature in the multi-horizon situation is
negative up to $r<r_{d}$ and then positive as $r>r_{d}$, as can be shown in
Fig. 3 2(a).

(a) The Hawking temperature associated with the horizon of the BH described in
Eq. (IV)

(b) The the Gibbs function of solution described in Eq. (IV)

Figure 3: Thermodynamic features of the BH solutions (IV) illustrated in
diagrams for parameter $v$ taking negative values: 2(a) characteristic pattern
of the black hole’s horizon Hawking temperature (IV) and 2(b) the heat
capacity of the black hole’s horizon (IV).

##  VIII Conclusions and discussion

The study of charged cylindrical black holes in the context of the
$f(\mathbb{Q})$ gravitational theory has yielded important discoveries about
these unusual astronomical objects. The research has included a thorough
investigation of how the $f(\mathbb{Q})$ change affects the characteristics
and actions of charged cylindrical black holes, providing insight into how
gravity and the modified theory interact.

The significant effect of the $f(\mathbb{Q})$ alteration on the geometric and
thermodynamic properties of charged cylindrical black holes is one of the main
conclusions of this work. The examination of thermodynamic parameters like
temperature, heat capacity, and entropy has uncovered fascinating trends that
have defied traditional wisdom and opened the door to a more comprehensive
comprehension of the underlying gravitational dynamics.

We give here a novel charged solution in the framework of the
Maxwell-$f(\mathbb{Q})$ gravitational theory for dimensions $N\geq 4$, with
the coincident gauge condition imposed Heisenberg (2023). For
$f(\mathbb{Q})=\mathbb{Q}+\frac{1}{2}\alpha\mathbb{Q}^{2}$, where $\alpha<0$,
we find a particular solution. Interesting aspects are shown by this solution,
which includes both quadrupole and monopole terms. The requirement that the
metric asymptotically approaches Anti-de Sitter (AdS) governs how these terms
interact.

Additionally, we explored the singularity innate to this solution and
clarified that there is a singularity at $r=0$ for any invariant derived from
the curvature and non-metricity. In comparison with the singularity of a
charged black hole in GR, this singularity is less severe. The invariants
asymptotic behavior is given as
$K=\mathbb{R}{\mu\nu}\mathbb{R}^{\mu\nu}=\mathbb{R}=\mathbb{Q}^{\alpha\beta\gamma}\mathbb{Q}{\alpha\beta\gamma}=\mathbb{Q}^{\alpha}\mathbb{Q}{\alpha}=\tilde{\mathbb{Q}}^{\alpha}\tilde{\mathbb{Q}}{\alpha}=\mathbb{Q}\sim
r^{-(N-2)}$, in opposition to the behavior seen in established solutions
within GR’s Einstein-Maxwell theory. Furthermore, the non-charged solution
obtained in this work, according to Eq. (III), shows characteristics in such a
way that $K=\mathbb{R}{\mu\nu}\mathbb{R}^{\mu\nu}\sim r^{-2N}$ and
$\mathbb{R}\sim r^{-N}$. Moreover, despite the fact that the charged solution
possesses distinct constituents of $g{tt}$ and $g^{rr}$, the event horizons
and Killing horizons of both cases coincide.

From a genuine physical perspective, these entities can contribute to the
current discussion on whether gravity model is the most reliable, be it
curvature or non-metricity picture. As explained in literature, since GR is
substantially equivalent, the discussion should center on $f(\mathbb{Q})$ and
$f(\mathbb{R})$ constructions because they are essentially different. Even
basic concepts such as gravitational waves show significant differences
between $f(\mathbb{Q})$ and $f(\mathbb{R})$ formulations, as Soudi et al.
(2019) illustrates. In resolving this dispute, a thorough grasp of black hole
properties may be extremely helpful.

## Acknowledgements

I would like to thank S. Nojiri for useful discussion when preparing this
manuscript and also I would like to thank Kobayashi-Moskawa institute for
their hosting while doing this study.

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# Groups With At Most Twelve Subgroups

Michael C. Slattery
Marquette University, Milwaukee WI 53201-1881
<EMAIL_ADDRESS>
In [2] I showed that the finite groups with a specified number of subgroups
can always be described as a finite list of similarity classes. Neil Sloane
suggested that I submit the corresponding sequence, the number of similarity
classes with $n$ subgroups, to his Online Encyclopedia of Integer Sequences
[4]. I thought this would be a quick calculation until I discovered that my
“example” case in the paper ($n=6$) was wrong (as noted in [3]). I realized
that producing a reliable count required a full proof. This note contains the
proof behind my computation of the first 12 terms of the sequence.

For completeness, we include a section of [2]:

> Now, we saw [above] that cyclic Sylow subgroups of $G$ which are direct
> factors allow many non-isomorphic groups to have the same number of
> subgroups. [This lemma] tells us that these are precisely the cyclic Sylow
> subgroups which lie in the center of $G$. Consequently, if
> $p_{1},\ldots,p_{c}$ are the primes which divide $|G|$ such that a Sylow
> $p_{i}$-subgroup is cyclic and central, then we can write
>
> $G=P_{1}\times P_{2}\times\cdots\times P_{c}\times O_{\pi^{\prime}}(G)$
>
> where each $P_{i}$ is a Sylow $p_{i}$-subgroup, the set
> $\pi=\\{p_{1},\ldots,p_{c}\\}$ and $O_{\pi^{\prime}}(G)$ is the largest
> normal subgroup of $G$ with order not divisible by any prime in $\pi$. We
> will write $\widetilde{G}=O_{\pi^{\prime}}(G)$. In other words,
> $\widetilde{G}$ is the unique subgroup of $G$ left after factoring out the
> cyclic, central Sylow subgroups. It is the part of $G$ that we can hope to
> restrict in terms of the number of subgroups. On the other hand, if we
> substituted a different prime (relatively prime to $|G|$) for any one of the
> $p_{i}$, it would not affect the number of subgroups of $G$. Thus, we are
> led to define the following equivalence relation (which we will call
> “similar” for this note).
>
> ######  Definition.
>
> Let $G$ and $H$ be two finite groups. Write $G$ as a product of cyclic
> central Sylow subgroups and $\widetilde{G}$ as above. Hence, $G=P_{1}\times
> P_{2}\times\cdots\times P_{c}\times\widetilde{G}$, and similarly,
> $H=Q_{1}\times Q_{2}\times\cdots\times Q_{d}\times\widetilde{H}$. We say
> that $G$ is similar to $H$ if, and only if, the following three conditions
> hold:
>
>   1. 1.
>
> $\widetilde{G}$ is isomorphic to $\widetilde{H}$;
>
>   2. 2.
>
> $c=d$;
>
>   3. 3.
>
> $n_{i}=m_{i}$ for some reordering, where $|P_{i}|=p_{i}^{n_{i}}$ and
> $|Q_{i}|=q_{i}^{m_{i}}$.
>
>

>
> From the comments before the definition, we see that if $G$ is similar to
> $H$, then they will have the same number of subgroups. Also, note that the
> equivalence class of $G$ is determined by $\widetilde{G}$ and the multiset
> $[n_{1},\ldots,n_{c}]$. If we have a bound on the number of subgroups of
> $G$, then this will bound $c$ and each $n_{i}$. Therefore, the only
> remaining hole in our theorem will be filled with the following lemma.
>
> ######  Lemma.
>
> Given $k>1$ there are a finite number of isomorphism classes of groups
> $G=\widetilde{G}$ (i.e. G has no cyclic, central Sylow subgroups) having at
> most $k$ subgroups.

We wish to identify all similarity classes of groups with at most 12
subgroups.

To begin, we will find all groups with $G=\widetilde{G}$ and at most 12
subgroups.

##  1 Abelian Groups

If $G$ is abelian, then no Sylow $p$-subgroup can be cyclic. This means that
if $G$ is not a $p$-group, then it has at least $p+1$ subgroups of order $p$
and $q+1$ subgroups of order $q$ for some distinct primes $p$ and $q$ and so
at least $(p+1)(q+1)\geq 12$ non-trivial subgroups. So, $G$ must be a non-
cyclic $p$-group.

If $G$ contains $Z_{p}\times Z_{p}\times Z_{p}$ then it has at least
$2p^{2}+2p+4\geq 16$ subgroups. Consequently, $G=Z_{p^{r}}\times Z_{p^{s}}$
with $r\geq s\geq 1$.

Now, $Z_{p^{2}}\times Z_{p^{2}}$ has $p^{2}+3p+5\geq 15$ subgroups, so we must
have $s=1$. Furthermore, $Z_{p^{4}}\times Z_{p}$ has $4p+6\geq 14$ subgroups,
so $r\leq 3$.

Hence, the abelian groups with $G=\widetilde{G}$ and at most 12 subgroups are:

$r=1$: ($p+3$ subgroups) p = 2, 3, 5, 7

$r=2$: ($2p+4$ subgroups) p = 2, 3

$r=3$: ($3p+5$ subgroups) p = 2

##  2 Non-Abelian Groups

Now we consider non-abelian $G$. Let $q$ be any prime dividing $|G|$. If Sylow
$q$ is not normal, there are at least $q+1$ Sylow $q$-subgroups. If Sylow $q$
is normal, there is a $q$-complement. If the complement is normal, then the
Sylow $q$ cannot be cyclic (else $G\neq\widetilde{G}$) and so has at least
$q+1$ maximal subgroups. Finally, if the complement is not normal, then we
have at least $q$ complements and 1 Sylow $q$. So, in every case, with the
trival subgroup and $G$, the number of subgroups is at least $q+3$. This
implies $q\leq 9$.

??? Can $|G|$ be div by all of 2, 3, 5, and 7 ??? As below, at least 2 must be
normal and their product has a complement with 2 primes that can have at most
3 subgroups $\Rightarrow\Leftarrow$ impossible.

Suppose $|G|$ is divisible by 3 distinct primes $p,q,r$.

If none of the Sylows are normal, we have at least $(p+1)+(q+1)+(r+1)\geq 13$
subgroups, so at least one is normal. Call it an $r$-subgroup $N$.
Complementing $N$, we have a two prime group $H$ in $G$ and every subgroup of
$H$ gives rise to at least two subgroups of $G$ (itself and itself times $N$).
So, $H$ has at most 6 subgroups.

???Must we have $H=\widetilde{H}$??? NO: $Z_{6}$ on $Z_{7}$.

######  Lemma.

A group $H$ divisible by 2 primes with at most 6 subgroups is either
$Z_{p}\times Z_{q},Z_{p}\times Z_{q^{2}}$, or $S_{3}$.

###### Proof.

If $H$ is abelian, one sylow has 2 subgroups, the other 2 or 3 subgroups, so
$H=Z_{p}\times Z_{q}$ or $Z_{p}\times Z_{q^{2}}$. If $H$ has a non-abelian
sylow, the sylow has at least 3 maximal subgroups, 1 prime order, 1 triv, 1
whole subgrp, plus the other sylow $\Rightarrow\geq 7$ subgrps, which is too
many. So both sylows of $H$ are abelian and so at least one is not normal.
Assume WLOG $p<q$. Then there are at least p+1 non-normal sylows, one other
sylow, the whole group, and trivial $\Rightarrow\geq p+4$ subgroups. So we
must have equality on our estimate which means, $p=2$, sylow 2 is not normal,
$p+1$ divides $|G|$, so $q=3$, sylow 3 is normal, sylows have no non-trivial
subgroups $\Rightarrow H=S_{3}$. ∎

If $H$ is abelian with cyclic sylows, then neither Sylow can act trivially on
$N$ or it would be central cyclic in $G$. Therefore $N$ cannot normalize the
sylows or $H$ and so the number of conjugates of each is at least $r$. This
gives at least $3r$ subgroups of $G$. Furthermore we have $G$, $N$, 1, and
$NH_{p}$ and $NH_{q}$, so at least $3r+5$ subgroups. Consequently, we must
have $r=2$.

Our list of subgroups includes only $N$ and 1 from subgroups of $N$. So, if
$N$ has at least 4 subgroups, then $G$ will have at least 13 subgroups.
Therefore, $N$ is cyclic of order 2 or 4. But odd primes cannot act non-
trivially on a cyclic group of order 2 or 4, so this situation is impossible.

Last, we consider the case of $H=S_{3}$. In this case, $r\geq 5$. If $H$ acts
trivially on $N$, then $N$ cannot be cyclic and so has at least 8 subgroups.
Since $S_{3}$ has 6 subgroups, $G$ will have at least 48 subgroups. So, $H$
acts non-trivially on $N$ and a sylow 2, $H_{2}$, must also act non-trivally
on $N$. In particular, $N$ does not normalize $H$ or $H_{2}$ and, as above,
this gives us at least $2r$ subgroups of $G$. We also have $G$, $N$, and 1.
Since $r$ is at least 5, this implies $G$ has more than 12 subgroups.

Consequently, $|G|$ must be divisible by at most two primes.

##  3 Groups with two primes in the order

Suppose $|G|=p^{a}q^{b}$, $p<q$ primes and $G$ is non-abelian.

The argument for two primes is more complicated than I would like. I’ll use
indentation to organize assumptions (like in computer code).

If neither sylow is cyclic,

then we have $(p+1)+(q+1)$ maximals in the sylows, the two sylows and the
whole group and trivial. That is, the number of subgroups is at least $p+q+6$.
This can only be less than or equal to 12 when $p=2,q=3$. Furthermore, the
close estimate means each sylow must be normal and prime squared order. That
forces $G$ to be abelian $\Rightarrow\Leftarrow$.

If sylow $p$ is cyclic,

If sylow $p$ is normal,

then the sylow $q$ cannot act non-trivially and so the sylow $p$ is cyclic and
central $\Rightarrow\Leftarrow$.

else sylow $p$ is not normal,

then the number of sylow $p$ is $\geq q$.

If sylow $q$ not cyclic,

we have at least $q+1$ maximals and so at least $2q+4$ subgroups. Again the
estimate forces sylow $q$ to be $Z_{3}\times Z_{3}$ and $p=2$, at least 10
subgroups. Furthermore, we cannot have 9 sylow 2’s, so there is a central
subgroup of order 3. Counting again, we have whole, trivial, one 9, four 3’s,
three sylow 2’s, three sylow 2 times central 3 $\Rightarrow\geq 13$ subgroups.
$\Rightarrow\Leftarrow$

else sylow $q$ is cyclic,

then $G$ must be supersolvable since both sylows are cyclic and it follows the
sylow $q$ must be normal. Sylow $p$ must act non-trivially on $q$, so
$(p,q)=(2,3),(2,5),(2,7)$, or $(3,7)$.
The only case where the action is not that of an element of order $p$ is when
$p=2,q=5$ when $Z_{4}$ acts faithfully on $Z_{5}$. In this case, $G$ will have
a quotient group of order 20 isomorphic to $Z_{4}$ acting faithfully on
$Z_{5}$. Since this group has 14 subgroups, we do not need to consider such
$G$. Therefore, we can assume the sylow $p$ acts as an element of order $p$.
The sylow $p$ fixes only the identity in sylow $q$ (only possible non-triv
action). This means the sylow $p$ is self-normalizing and so the number of
sylow $p$ subgroups is $q^{b}$. Now the subgroup of order $p^{a-1}$
centralizes the sylow $q$ and so is the intersection of the sylow $p$’s. Hence
the non-triv $p$-subgroups are order $p,\ldots,p^{a-1}$, and $q^{b}$ subgroups
order $p^{a}$. There are $b$ non-triv $q$-subgroups and $(a-1)\cdot b$ proper
abelian subgroups divisible by $pq$. Furthermore, for each non-triv, proper
$q$-subgroup, order $q^{c}$, we can multiply that by the sylow $p$’s to get
subgroups of order $p^{a}q^{c}$. There will be $q^{c}$ different sylow $p$’s
giving the same subgroup of order $p^{a}q^{c}$ and so a total of $q^{b-c}$
such subgroups. Consequently, the number of (non-abelian) proper subgroups
divisible by $p^{a}q$ will be $q+q^{2}+\cdots+q^{b-1}=(q^{b}-q)/(q-1)$.
Including trivial and whole group, the number of subgroups is:

$(a-1+q^{b})+b+(a-1)b+\frac{q^{b}-q}{q-1}+2=q^{b}+\frac{q^{b}-q}{q-1}+a(b+1)+1.$

The corresponding values are shown in the following tables.

$q=3$
---
| $b$
|  | 1 | 2
| 1 | 6 | 16
$a$ | 2 | 8 | 19
| 3 | 10 | 22
| 4 | 12 | 25
| 5 | 14 | 28
$q=5$
---
| $b$
|  | 1 | 2
| 1 | 8 | 34
$a$ | 2 | 10 | 37
| 3 | 12 | 40
| 4 | 14 | 43
$q=7$
---
| $b$
|  | 1 | 2
| 1 | 10 | 60
$a$ | 2 | 12 | 63
| 3 | 14 | 66

Since the $q=7$ table applies to both $p=2$ and $p=3$ we see there are 11
groups with at most 12 subgroups in this case.

else sylow $p$ is not cyclic,

and so sylow $q$ must be cyclic and not central.

If neither sylow is normal,

then number of sylow $p$ is at least $q$ and the number of sylow $q$ is at
least $q+1$. Including trivial and whole, we have at least $2q+3$ subgroups,
which means $q<5$ and so we have $q=3,p=2$. We have at least three sylow 2’s
and at least four sylow 3’s, triv, whole, and one sylow 2 will contain three
subgroups of order 2. However, that’s already 12 subgroups and we have more
order 2 subgroups in the other sylow 2’s. $\Rightarrow\Leftarrow$

If sylow $p$ is normal,

then it cannot be cyclic, else it would be central. Furthermore, $Z_{p}\times
Z_{p}\times Z_{p}$ has too many subgroups, so the sylow $p$ must be a two-
generator group. So, the sylow $p$ has $p+1$ maximal subgroups, there are at
least $q+1$ sylow $q$’s and with sylow $p$ itself, triv, and whole, we have at
least $p+q+5$ subgroups. If $p\geq 3$, this is too many, so we must have
$p=2$. If $q\geq 5$, then our count gives at least 12 subgroups. The only way
to avoid going over 12 would be to have the sylow $p$ be $Z_{2}\times Z_{2}$,
sylow $q$ be $Z_{5}$. But $Z_{5}$ has no non-trivial action on $Z_{2}\times
Z_{2}$, and so we are left only with the case $p=2,q=3$, and at least 10
subgroups. Now, the alternating group $A_{4}$ satisfies these conditions and
has 10 subgroups. If sylow 3 has order larger than 3, then there will be a
central 3-subgroup whose product with the various 2-subgroups will give more
than 12 subgroups. So, the sylow 3 must be order 3. If sylow 2 has order
larger than 4, then the frattini subgroup of the sylow 2 and the frattini
subgroup times the various sylow 3-subgroups give more than 12 subgroups. So,
$A_{4}$ is the only group in this case.

If sylow $q$ is normal,

we are already assuming the sylow $q$ is cyclic. Furthermore, the sylow $p$ is
not normal and not cyclic. So we have at least $q$ sylow $p$’s and the sylow
$p$’s have at least $p+1$ maximal subgroups.

If two sylow $p$’s have a common maximal subgroup, $M$,

then the normalizer of $M$ will contain at least two sylow $p$-subgroups of
$G$ and so must be divisible by $q$. Thus the intersection of the normalizer
with the sylow $q$-subgroup is a non-trivial central $q$ subgroup. Therefore,
our subgroups include $q$ sylow $p$’s, $q$ sylow $p$’s times the central $q$,
at least $p+1$ maximal subgroups of sylow $p$, each of those $p+1$ times the
central $q$, the sylow $q$, the central $q$, triv, and whole for at least
$2q+2p+6\geq 16$ subgroups. $\Rightarrow\Leftarrow$

else no shared maximal subgroups,

which means we have at least $q(p+1)$ maximal subgroups of sylow $p$’s
altogether. Therefore we have at least these $q(p+1)$ subgroups, $q$ sylow
$p$-subgroups, one sylow $q$, triv, and whole. That is, $qp+2q+3\geq 15$
subgroups. $\Rightarrow\Leftarrow$

Hence, the non-abelian groups with $G=\widetilde{G}$, $|G|$ divisible by at
least two primes, and at most 12 subgroups are:

$Z_{p}\ltimes Z_{q}$: ($q+3$ subgroups) $(p,q)$ = (2, 3), (2, 5), (2, 7), or
(3, 7)

$Z_{p^{2}}\ltimes Z_{q}$ with action of order $p$: ($q+5$ subgroups) $(p,q)$ =
(2, 3), (2, 5), (2, 7), or (3, 7)

$Z_{8}\ltimes Z_{q}$ with action of order 2: ($q+7$ subgroups) $q$ = 3 or 5

$Z_{16}\ltimes Z_{3}$ with action of order 2: (12 subgroups)

$A_{4}$: (10 subgroups)

##  4 $p$-Groups

Finally, we have the case of $G$ a non-abelian $p$-group, $|G|=p^{n}$.

The group $Z_{p}\times Z_{p}\times Z_{p}$ has at least 16 subgroups, so $G$
must be a two-generator group. Consequently, we have $G$, $p+1$ maximals, the
Frattini subgroup, and triv $\Rightarrow\geq p+4$ subgroups.

When $p$ is odd we will have at least $p+1$ order $p$, one of which might be
the Frattini, so we’d have at least $2p+4\Rightarrow p$ = 2 or 3.

Assume $|G^{\prime}|=p$ and $Z(G)$ cyclic. Then $G^{\prime}\subset Z(G)$ and
for $x,y\in G$, $[x^{p},y]=[x,y]^{p}=1$ so $x^{p}$ is central and
$\Phi(G)\subset Z(G)$. Now $|G:\Phi(G)|=p^{2}$ and so we must have
$\Phi(G)=Z(G)$ is cyclic order $p^{n-2}$.

If $G$ has a single subgroup of order $p$, then $p=2$ and $G$ is generalized
quaternion. Now, each generalized quaternion 2-group contains the next smaller
as a subgroup. So, we can just check: $Q_{8}$ has 6 subgroups, $Q_{16}$ has 11
subgroups, and $Q_{32}$ has 20. That is, $G$ must be $Q_{8}$ or $Q_{16}$.

Otherwise, $G$ has more than one subgroup of order $p$ and so we can choose
$S\subset G$ with $S$ order p and not in the center. Then $M=SZ(G)$ is an
abelian subgroup isomorphic to $Z_{p}\times Z_{p^{n-2}}$. From the abelian
case above, we see that the number of subgroups of $M$ is given in the
following table.

| $n$
---|---
|  | 3 | 4 | 5
$p$ | 2 | 5 | 8 | 11
| 3 | 6 | 10 | 14

In addition to the subgroups in $M$, $G$ also has $p$ other maximal subgroups,
and $G$ itself. So, increasing each table entry by $p+1$ we see the only
possibilities for at most 12 subgroups are $|G|$ = 8, 16, or 27.

By brute force check we find for order 8 the dihedral group $D_{8}$ with 10
subgroups, for order 16 a group with presentation $\langle
a,b|a^{2},b^{8},b^{a}=b^{5}\rangle$ having 11 subgroups, and for order 27 the
extraspecial group of exponent 9 with 10 subgroups.

Thus we have found five groups with $|G^{\prime}|=p$ and cyclic center. Now
any non-abelian finite $p$-group will have such a group as a homomorphic
image.

Note that no generalized quaternion group can have a smaller generalized
quaternion group as a homomorphic image. One easy way to see this is to note
that $Q_{n}/Z(Q_{n})=D_{n/2}$ and any non-abelian image of a dihedral group
will have more than one involution. We will use this fact several times below.

Suppose $|G|=3^{5}$ and choose $K$ maximal such that $\overline{G}=G/K$ is
non-abelian. It follows that $\overline{G}$ will have
$|\overline{G}^{\prime}|=3$ and $Z(\overline{G})$ cyclic. If $|K|\leq 3$, it
follows from above that $\overline{G}$, and so $G$, will have more than 12
subgroups. If $|K|=3^{2}$, then $|\overline{G}|=3^{3}$ and so must be the
extraspecial group of exponent $3^{2}$. Thus, $\overline{G}$ has 10 subgroups.
The only way we could have at most 12 subgroups in $G$ is if $K$ is cyclic and
every subgroup either contains $K$ or is order 3 or 1 in $K$. However, this
implies that $G$ has only one subgroup of order 3, which is impossible.
Consequently, no non-abelian 3-group of order at least $3^{5}$ can have at
most 12 subgroups. We know from above that there is only one such group of
order $3^{3}$ and a computer check shows that there are no examples of order
$3^{4}$ (these groups have at least 14 subgroups).

Now suppose $|G|=2^{6}$ and choose $K$ maximal such that $\overline{G}=G/K$ is
non-abelian. As above, $\overline{G}$ will have $|\overline{G}^{\prime}|=2$
and $Z(\overline{G})$ cyclic. If $|K|\leq 2$, it follows from above that
$\overline{G}$, and so $G$, will have more than 12 subgroups. If $|K|=2^{2}$,
then $|\overline{G}|=2^{4}$ and so we see from above that $\overline{G}$ is
one of two groups each of which have 11 subgroups. Since $K$ has at least 2
proper subgroups, $G$ must have 13 or more subgroups. Finally, consider
$|K|=2^{3}$. Then $\overline{G}$ must be $D_{8}$ or $Q_{8}$ with 10 or 6
subgroups respectively. Since $K$ has at least 3 proper subgroups,
$\overline{G}$ cannot be $D_{8}$ and so must be $Q_{8}$ with 6 subgroups. Even
if all of the subgroups of $G$ either contained $K$ or were contained in $K$,
then $K$ would have to have at most 7 subgroups (6 proper). However, the only
groups of order $2^{3}$ with 7 or fewer subgroups are $Z_{8}$ and $Q_{8}$.
Since each of these have a single subgroup of order 2, our assumption would
force $G$ to be generalized quaternion, which it clearly is not. Thus we see
that no non-abelian 2-group of order at least $2^{6}$ can have number of
subgroups less than or equal to 12. A computer check shows there are no
examples of order $2^{5}$ (these groups have at least 14 subgroups) and only
the two groups mentioned above for order $2^{4}$.

Hence, the non-abelian $p$-groups with $G=\widetilde{G}$ and at most 12
subgroups are:

$D_{8}$: (10 subgroups)

$Q_{8}$: (6 subgroups)

$Q_{16}$: (11 subgroups)

$Z_{2}\ltimes Z_{8}$: (11 subgroups)

$E_{27}$ extraspecial order 27, exponent 9: (10 subgroups)

##  5 Conclusion

Collecting all of our results, we have:

$n$ | Groups with $G=\widetilde{G}$ and $n$ subgroups
---|---
1 | Trivial group
2 |
3 |
4 |
5 | $Z_{2}\times Z_{2}$
6 |  $Z_{3}\times Z_{3}$, $S_{3}$, $Q_{8}$
7 |
8 |  $Z_{2}\times Z_{4}$, $Z_{5}\times Z_{5}$, $D_{10}$, $Z_{4}\ltimes Z_{3}$
9 |
10 |  $Z_{3}\times Z_{9}$, $Z_{7}\times Z_{7}$, $D_{14}$, $A_{4}$, $Z_{3}\ltimes Z_{7}$, $Z_{4}\ltimes Z_{5}$, $Z_{8}\ltimes Z_{3}$, $E_{27}$
11 |  $Z_{2}\times Z_{8}$, $Q_{16}$, $Z_{2}\ltimes Z_{8}$
12 |  $Z_{4}\ltimes Z_{7}$, $Z_{9}\ltimes Z_{7}$, $Z_{8}\ltimes Z_{5}$, $Z_{16}\ltimes Z_{3}$

In particular the sequence of the number of groups with $G=\widetilde{G}$ and
$n$ subgroups would be:

$1,0,0,0,1,3,0,4,0,8,3,4,\ldots$

Forming the direct product with a coprime, cyclic group of order $p^{k}$ will
multiply the number of subgroups by $k+1$. Thus we find groups with $n$
subgroups corresponding to various factorizations of $n$.

$n$ | Similarity class representatives with $n$ subgroups
---|---
1 | Trivial group
2 | $Z_{2}$
3 | $Z_{2^{2}}$
4 |  $Z_{2^{3}}$, $Z_{2}\times Z_{3}$
5 |  $Z_{2^{4}}$, $Z_{2}\times Z_{2}$
6 |  $Z_{2^{5}}$, $Z_{2}\times Z_{3^{2}}$, $Z_{3}\times Z_{3}$, $S_{3}$, $Q_{8}$
7 | $Z_{2^{6}}$
8 |  $Z_{2^{7}}$, $Z_{2}\times Z_{3^{3}}$, $Z_{2}\times Z_{3}\times Z_{5}$, $Z_{2}\times Z_{4}$, $Z_{5}\times Z_{5}$, $D_{10}$, $Z_{4}\ltimes Z_{3}$
9 |  $Z_{2^{8}}$, $Z_{2^{2}}\times Z_{3^{2}}$
10 |  $Z_{2^{9}}$, $Z_{2}\times Z_{3^{4}}$, $Z_{2}\times Z_{2}\times Z_{3}$, $Z_{3}\times Z_{9}$, $Z_{7}\times Z_{7}$, $D_{14}$, $A_{4}$, $Z_{3}\ltimes Z_{7}$, $Z_{4}\ltimes Z_{5}$, $Z_{8}\ltimes Z_{3}$, $E_{27}$
11 |  $Z_{2^{10}}$, $Z_{2}\times Z_{8}$, $Q_{16}$, $Z_{2}\ltimes Z_{8}$
12 |  $Z_{2^{11}}$, $Z_{2}\times Z_{3^{5}}$, $Z_{3}\times Z_{3}\times Z_{2}$, $S_{3}\times Z_{5}$, $Q_{8}\times Z_{3}$, $Z_{2^{2}}\times Z_{3^{3}}$, $Z_{2}\times Z_{3}\times Z_{5^{2}}$, $Z_{4}\ltimes Z_{7}$,
|  $Z_{9}\ltimes Z_{7}$, $Z_{8}\ltimes Z_{5}$, $Z_{16}\ltimes Z_{3}$

In the following version of the previous table we represent classes using
$p,q,r$ to represent primes which do not occur anywhere else in the group
order. This makes it a bit easier to recognize the infinite classes.

$n$ | Similarity classes with $n$ subgroups
---|---
1 | Trivial group
2 | $Z_{p}$
3 | $Z_{p^{2}}$
4 |  $Z_{p^{3}}$, $Z_{p}\times Z_{q}$
5 |  $Z_{p^{4}}$, $Z_{2}\times Z_{2}$
6 |  $Z_{p^{5}}$, $Z_{p}\times Z_{q^{2}}$, $Z_{3}\times Z_{3}$, $S_{3}$, $Q_{8}$
7 | $Z_{p^{6}}$
8 |  $Z_{p^{7}}$, $Z_{p}\times Z_{q^{3}}$, $Z_{p}\times Z_{q}\times Z_{r}$, $Z_{2}\times Z_{4}$, $Z_{5}\times Z_{5}$, $D_{10}$, $Z_{4}\ltimes Z_{3}$
9 |  $Z_{p^{8}}$, $Z_{p^{2}}\times Z_{q^{2}}$
10 |  $Z_{p^{9}}$, $Z_{p}\times Z_{q^{4}}$, $Z_{2}\times Z_{2}\times Z_{p}$, $Z_{3}\times Z_{9}$, $Z_{7}\times Z_{7}$, $D_{14}$, $A_{4}$, $Z_{3}\ltimes Z_{7}$, $Z_{4}\ltimes Z_{5}$, $Z_{8}\ltimes Z_{3}$, $E_{27}$
11 |  $Z_{p^{10}}$, $Z_{2}\times Z_{8}$, $Q_{16}$, $Z_{2}\ltimes Z_{8}$
12 |  $Z_{p^{11}}$, $Z_{p}\times Z_{q^{5}}$, $Z_{3}\times Z_{3}\times Z_{p}$, $S_{3}\times Z_{p}$, $Q_{8}\times Z_{p}$, $Z_{p^{2}}\times Z_{q^{3}}$, $Z_{p}\times Z_{q}\times Z_{r^{2}}$, $Z_{4}\ltimes Z_{7}$,
|  $Z_{9}\ltimes Z_{7}$, $Z_{8}\ltimes Z_{5}$, $Z_{16}\ltimes Z_{3}$

In conclusion, the sequence of number of similarity classes with a given
number of subgroups begins:

$1,1,1,2,2,5,1,7,2,11,4,11,\ldots$

Note: In [1], Miller lists the groups with specified number of subgroups where
the number of subgroups runs from 1 through 9. His lists agree with ours.

## References

  * [1] G.A. Miller, Groups having a small number of subgroups, Proc. Natl. Acad. Sci. U S A, vol. 25 (1939) 367–371.
  * [2] M.C. Slattery, On a property motivated by groups with a specified number of subgroups, Amer. Math. Monthly, vol. 123 (2016) 78–81.
  * [3] M.C. Slattery, Editor’s End Notes, Amer. Math. Monthly, vol. 123 (2016) 515.
  * [4] N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org.