# Sakaguchi Swarmalators

Joao U.F. Lizárraga  Marcus A.M. de Aguiar<EMAIL_ADDRESS>Instituto de
Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp 13083-970,
Campinas, São Paulo, Brazil

###### Abstract

Swarmalators are phase oscillators that cluster in space, like fireflies
flashing on a swarm to attract mates. Interactions between particles, which
tend to synchronize their phases and align their motion, decrease with the
distance and phase difference between them, coupling the spatial and phase
dynamics. In this work, we explore the effects of disorder induced by phase
frustration on a system of Swarmalators that move on a one-dimensional ring.
Our model is inspired by the well-known Kuramoto-Sakaguchi equations. We find,
numerically and analytically, the ordered and disordered states that emerge in
the system. The active states, not present in the model without disorder,
resemble states found previously in numerical studies for the 2D Swarmalators
system. One of these states, in particular, shows similarities to turbulence
generated in a flattened media. We show that all ordered states can be
generated for any values of the coupling constants by tuning the phase
frustration parameters only. Moreover, many of these combinations display
multi-stability.

##  I Introduction

Synchronization and swarming are emergent phenomena observed in various living
systems. The former refers to the tendency of individuals’ states to converge
towards specific periodic behaviors and has been widely investigated using the
Kuramoto [1, 2] or the Stuart-Landau [3, 4] models. The latter describes
systems in which individuals tend to aggregate and align in space, as often
observed in animals such as birds and fish [5, 6]. Although the two behaviors
have been spotted independently in nature, systems including the Japanese tree
frogs [7] and the Quincke rollers [8], among others [9, 10, 11], suggest that
synchronization and swarming also occur together. A model that couples both
behaviors was recently proposed in [12] and the corresponding particles termed
swarmalators.

The Swarmalators model [12] describes a system of particles characterized by
internal phases $\theta_{i}$ and spatial positions $\vec{x}_{i}$. Phase and
position dynamics coupled in such a way that phases tend to synchronize among
nearby particles and velocities tend to align more easily among particles with
synchronized phases. An instance of the model, for $N$ particles moving in a
two-dimensional space, is described by

$\displaystyle\begin{split}\dot{\vec{x}}_{i}&=\frac{1}{N}\sum_{j\neq
i}^{N}\bigg{[}\frac{\vec{x}_{j}-\vec{x}_{i}}{|{\vec{x}_{j}-\vec{x}_{i}}|}(1+J\cos(\theta_{j}-\theta_{i}))-\frac{\vec{x}_{j}-\vec{x}_{i}}{|{\vec{x}_{j}-\vec{x}_{i}}|^{2}}\bigg{]}\\\
\dot{\theta}_{i}&=\frac{K}{N}\sum_{j\neq
i}^{N}\frac{\sin(\theta_{j}-\theta_{i})}{|\vec{x}_{j}-\vec{x}_{i}|}.\end{split}$
(1)

It has been shown that different collective states may emerge for specific
sets of parameters $K$ and $J$ [12]. Previous work have also explored the
system’s behavior under external stimulus [13], variations on the nature of
individual’s interactions [14, 15, 16, 17, 18] and effects of thermal noise
[19]. However, from an analytical perspective almost no progress has been
made. Under this premise, O’Keeffe et. al [20] proposed a one-dimensional
analogue of the model whose dynamics are governed by

$\displaystyle\begin{split}\dot{x_{i}}=\frac{J}{N}\sum_{j}\sin(x_{j}-x_{i})\cos(\theta_{j}-\theta_{i}),\\\
\dot{\theta_{i}}=\frac{K}{N}\sum_{j}\sin(\theta_{j}-\theta_{i})\cos(x_{j}-x_{i}),\end{split}$
(2)

and capture several features of Eqs. (1). This simpler model displays the
emergence of several static collective states observed in the 2D system and,
most importantly, can be treated analytically. Further work have also
considered noisy interactions [19], distributed couplings [21], random pinning
[22], and intrinsic oscillating frequencies [23].

The 1D Swarmalators model, however, cannot describe some of the active states
displayed by the full 2D system. Even if some states of the 2D model are
arranged in the form of an annulus, projecting it onto a 1D ring leaves out
part of the dynamics that could be essential for the formation of the
structure. On the other hand, the similarity of the 1D model with a pair of
coupled Kuramoto equations, suggests that the expertise acquired from studying
this famous synchronization model can be leveraged to analyze Swarmalators
systems of this type. Here we propose a model of frustrated 1D swarmalators,
based on the Sakaguchi-Kuramoto model [24], as a source of disorder that could
compensate for the loss of freedom of the 1D system and potentially restore
the active states observed in the 2D model. We call the corresponding
particles Sakaguchi Swarmalators. We explore the effects of additional phase
frustration parameters to both the spatial and phase dynamics in Eqs. (2).
This type of disorder differs from (and complements) that produced by
distributed couplings, studied by several authors for the Kuramoto model [1],
and by O’Keeffe and Hong for the 1D Swarmalators model [21]. We will show
that, indeed, frustration leads to coherent active states in 1D, similar to
the ones found in the 2D setup. Additionally, it is worth mentioning that the
new parameters of the Sakaguchi Swarmalators model have similarity with
“offset terms” used in a recent numerical study of a modified 2D Swarmalators
system [25].

We describe the modifications to the original 1D model in (Sec. II),
introducing disorder as in the Kuramoto-Sakaguchi model. Then, in the same
section, we present the different collective states obtained from numerical
computations. In Sec. III, we present the stability analyses of states that
show ordered configurations. The conditions obtained from the analytical
computations allow us to picture the stability regions in the space of
frustration parameters (Sec. IV). Finally, in Sec. V we sum up some concluding
remarks.

##  II The Sakaguchi Swarmalators model

Our modifications to the 1D Swarmalators model bring back the essential
feature of the Kuramoto-Sakaguchi model: the introduction of disorder on a
system intended to exhibit a coherent behavior [24]. In our model, however,
the disorder affects both the spatial and phase components of each particle’s
dynamics. As described by the expressions

$\displaystyle\begin{split}\dot{x_{i}}=\frac{J}{N}\sum_{j}\sin(x_{j}-x_{i}+\alpha)\cos(\theta_{j}-\theta_{i}+\beta),\\\
\dot{\theta_{i}}=\frac{K}{N}\sum_{j}\sin(\theta_{j}-\theta_{i}+\beta)\cos(x_{j}-x_{i}+\alpha)\end{split}$
(3)

so that the disorder parameters $\alpha$ and $\beta$ are incorporated to the
system dynamics. Hence, if the system reaches coherence in phase
($\theta_{j}\approx\theta_{i}$) or space ($x_{j}\approx x_{i}$), the effects
induced by $\alpha$ and $\beta$ would lead to its disruption.

###  II.1 Equilibrium states

Numerical computation of Eqs. (3) allows us to get insights on its long term
behavior. Snapshots of the system’s collective behavior, after $10^{4}$ time
steps, are shown in Figures 1, 2, and 3. In all cases, the number of particles
is $N=500$ and the parameter $J$ is set to $1$. Thus, the control parameters
are $K$, $\alpha$, and $\beta$. In the figures we use the auxiliary parameters
$\gamma^{\pm}=\alpha\pm\beta$, instead, since $\gamma^{\pm}$ will be relevant
for the stability analyses in the next sections. Particles’ positions and
phases are initially distributed uniformly in ranges $-\pi$ to $\pi$. In the
spatial pictures, shown in the top rows of Figures 1, 2, and 3, particles are
positioned along the ring and colored according to their phases. The scatter
plots, in the bottom row of Figures 1, 2, and 3, correspond to the spatial
picture above and show the position-phase correlation for each particle. The
states presented in Figure 1 demonstrate the convergence of particles to fixed
values in phase and space. Once the particles reach these states, after a
transient period, they remain there statically. Despite this feature, however,
we observe clear differences in the position-phase coherence of each case. The
Static Synchronous state (Figures 1a and 1d) shows the formation of two
clusters spaced, in phase and space, by a factor of $\pi$. Particles move to
each cluster depending on their initial condition and synchronize with its
cluster neighbors. In the Static Phase Wave state, (Figures 1b and 1e),
particles are uniformly distributed along the ring. Moreover, each particle’s
phase is correlated with its position, implying that these are also
distributed uniformly. The correlation shown in the figure is positive,
however, variations in the initial conditions can change the behavior of the
system so that the steady behavior leads to a negative correlation. The Static
Asynchronous state (Figures 1c and 1d) shows that particles are distributed
uniformly in phase and space. However, unlike the Static Phase Wave state,
particles’ positions and phases are uncorrelated.

Figure 1: Spatial behavior (top row) and position-phase correlations (bottom
row) for the static states of the Sakaguchi Swarmalators model. ($K=1$) is set
for all the simulations, and ($\gamma^{+}$, $\gamma^{-}$) are set as a,d)
($0$, $0$) for the Static Synchronous, b,e) ($1.67$, $0$) for the Static Phase
Wave, and c,f) ($1.8$, $2$) for the Static Asynchronous states.

Active analogs of the Static Synchronous and Static Phase Wave states are
shown in Figure 2. In the Active Synchronous state (Figures 2a and 2d) the two
clusters of particles, as described before for the Static Synchronous case,
are rotating along the ring. Despite the rotation, the clusters preserve the
spacing of $\pi$ in position and phase. A similar effect is seen in the Active
Phase Wave state (Figures 2b and 2e), where the uniformily distributed
particles rotate while keeping the position-phase correlation. In Figures 2c
and 2f, we introduce a new state where particles rotate around the ring while
keeping a fixed pattern. In this state, particles cluster on a position-phase
region, in contrast to the Active Synchronous state, where particles cluster
on two $\pi$-distanced points.

Figure 2: Spatial behavior (top row) and position-phase correlations (bottom
row) for the ordered active states of the Sakaguchi Swarmalators model. ($K$,
$\gamma^{+}$, $\gamma^{-}$) are set as a,d) ($1$, $1.3$, $1.5$) for the Active
Synchronous, b,e) ($1$, $2.5$, $1.1$) for the Active Phase Wave, and c,f)
($-5$, $0.3$, $-3.1$) for the Ring states (see movies S1, S2, and S3 in
Supplemental Material). Arrows represent the translation direction of the
particles.

In Figure 3, we show three additional active states. Despite not being
completely ordered these states still show the emergence of intriguing
patterns. In the Noisy Active Phase Wave state (Figures 3a and 3d), particles
move and initially form a correlated position-phase pattern. After some time,
however, this coherence is destroyed and a dynamic behavior starts where
distorted correlation appears and disappears continuously. The Active
Asynchronous state (Figures 3b and 3e) is the active analogous to the Static
Asynchronous state, shown in Figures 2c and 2e. In this state, however,
particles jiggle and move randomly. The effects of this dynamic behavior, as
better shown in the state’s scatter plot, generate position-phase correlation
in a non-uniformly distributed configuration. The last disordered state
(Figures 3c and 3f) is named Turbulent and is unrelated to the previous ones.
In this state, the particles move randomly along the ring without an specific
position-phase coherence. However, as presented on its scatter plot, a
recurrent pattern emerges where the particles’ position-phase correlations
generate vortexes that rotate and move around while exchanging individuals.

Figure 3: Spatial behavior (top row) and position-phase correlations (bottom
row) for the disordered active states of the Sakaguchi Swarmalators model.
($K$, $\gamma^{+}$, $\gamma^{-}$) are set as a,d) ($-0.2$, $1.9$, $-1.4$) for
the Noisy Active Phase Wave, b,e) ($-1$, $1.25$, $-0.25$) for the Active
Asynchronous, and c,f) ($-0.2$, $3$, $-0.5$) for the Turbulent states (see
movies S4, S5, and S6 in Supplemental Material). In all cases, particles are
in continuous motion.

The figures described in this section, obtained numerically, allow us to
understand the coherent and incoherent behaviors of the Sakaguchi Swarmalators
system. In the next section we explore the analytical features of the model,
describing some of these states and their stability conditions.

##  III Stability analyses

Following [20] we define $\xi_{i}=x_{i}+\theta_{i}$,
$\eta_{i}=x_{i}-\theta_{i}$ and rewrite Eqs. (3) as

$\displaystyle\begin{split}\dot{\xi}_{i}&=\frac{J_{+}}{N}\sum_{j}\sin(\xi_{j}-\xi_{i}+\gamma^{+})+\frac{J_{-}}{N}\sum_{j}\sin(\eta_{j}-\eta_{i}+\gamma^{-}),\\\
\dot{\eta}_{i}&=\frac{J_{-}}{N}\sum_{j}\sin(\xi_{j}-\xi_{i}+\gamma^{+})+\frac{J_{+}}{N}\sum_{j}\sin(\eta_{j}-\eta_{i}+\gamma^{-}),\end{split}$
(4)

where $J_{\pm}=(J\pm K)/2$ and $\gamma^{\pm}=\alpha\pm\beta$. We also define
the order parameters

$\displaystyle\begin{split}S_{+}e^{\mathrm{i}\mkern
1.0mu\phi_{+}}&=\frac{1}{N}\sum_{j}e^{\mathrm{i}\mkern 1.0mu\xi_{j}},\\\
S_{-}e^{\mathrm{i}\mkern
1.0mu\phi_{-}}&=\frac{1}{N}\sum_{j}e^{\mathrm{i}\mkern
1.0mu\eta_{j}},\end{split}$ (5)

where the real values $S_{\pm}$ (ranging from $0$ to $1$) are coherence
metrics associated with positive or negative correlations between particles’
positions and phases. For instance, the coherence of the Static Phase Wave
state, shown in Figure 1b, is $S_{+}\approx 1$ due to the position-phase
positive correlation of the particles. For the Static Asynchronous state,
shown in Figure 1c, on the other hand, $S_{\pm}\approx 0$, since there is no
correlation between particles’ positions and phases.

###  III.1 Synchronous states

These states involve clustering and synchronization of the particles, that
converge simultaneously to specific values in phase and space that can be
static or dynamic. Setting $\xi_{j}=\xi_{i}=\xi$ and $\eta_{j}=\eta_{i}=\eta$
in Eqs. (4) we obtain the equilibrium trajectories

$\displaystyle\xi$
$\displaystyle=J_{+}\sin(\gamma^{+})t+J_{-}\sin(\gamma^{-})t+\xi_{0},$
$\displaystyle\eta$
$\displaystyle=J_{-}\sin(\gamma^{+})t+J_{+}\sin(\gamma^{-})t+\eta_{0}.$

To analyze the stability of this solution, we add small perturbations
$\delta\xi_{i}$ and $\delta\eta_{i}$ to each particle around the equilibrium
trajectory and compute their dynamic behavior. The temporal evolution of the
perturbations is described by

$\displaystyle\begin{split}\delta\dot{\xi_{i}}&=J^{+}\cos(\gamma^{+})\sum_{j}(\delta\xi_{j}-\delta\xi_{i})+J^{-}\cos(\gamma^{-})\sum_{j}(\delta\eta_{j}-\delta\eta_{i})\\\
\delta\dot{\eta_{i}}&=J^{-}\cos(\gamma^{+})\sum_{j}(\delta\xi_{j}-\delta\xi_{i})+J^{+}\cos(\gamma^{-})\sum_{j}(\delta\eta_{j}-\delta\eta_{i}),\end{split}$
(6)

where $J^{\pm}=J_{\pm}/N$. These equations form a $2N\times 2N$ linear system
which is evaluated in detail in Appendix A. The eigenvalues, that determine
the stability of the equilibrium trajectory, are

$\displaystyle\begin{split}\lambda_{\pm}^{SS}&=-\frac{J_{+}}{2}\left(\cos({\gamma^{+}})+\cos({\gamma^{-}})\right)\pm\frac{1}{2}\left({J_{+}}^{2}\left(\cos({\gamma^{+}})-\cos({\gamma^{-}})\right)^{2}+4{J_{-}}^{2}\cos({\gamma^{+}})\cos({\gamma^{-}})\right)^{1/2},\\\
\lambda_{0}^{SS}&=0,\end{split}$ (7)

where $\lambda_{\pm}^{SS}$ have multiplicity $(N-1)$ each, and
$\lambda_{0}^{SS}$ has multiplicity $2$. The superscript $SS$ stands for
Synchronous States.

For the particular case where $J=K$, and therefore $J_{-}=0$, the non-zero
eigenvalues are

$\displaystyle\begin{split}\lambda_{+}^{SS(J=K)}&=-J_{+}\cos(\gamma^{-}),\\\
\lambda_{-}^{SS(J=K)}&=-J_{+}\cos(\gamma^{+}),\end{split}$ (8)

and will be negative for $J_{+}>0$ when $\gamma^{\pm}\in[-\pi/2,\pi/2]$, and
for $J_{+}<0$ when $\gamma^{\pm}\in[\pi/2,3\pi/2]$. The stability regions for
$J\neq K$ are more complicated due to the shape of the non-zero eigenvalues
surfaces. We shown the stability regions in this case in the next section.

Before we close this subsection we note an interesting symmetry that appears
for $J=1$. In this specific case, $J_{\pm}=(K\pm 1)/2$, and considering
$K^{*}=1/K$, we get $J_{\pm}^{*}=\pm J_{\pm}/K$. Then, if we find
${J_{\pm}^{*}}^{2}$ an plug it into Eqs. (7), we see that the non-zero
eigenvalues will just be scaled as

${\lambda_{\pm}^{SS}}^{*}=\frac{1}{K}\lambda_{\pm}^{SS},$ (9)

so the stability regions for $K$ and $1/K$ are exactly the same.

###  III.2 Phase Wave states

Here the particles are distributed uniformly in space and phase but these
variables are correlated. Also, they can move rigidly, keeping their relative
positions and phases constant. These states are represented by
$x_{i}=2i\pi/N+x_{0}+v_{x}t$ and $\theta_{i}=\pm
2i\pi/N+\theta_{0}+v_{\theta}t$, where $v_{x}$ and $v_{\theta}$ can be
determined from Eqs. (4). The $\pm$ sign in $\theta_{i}$ depends on the type
of position-phase correlation. We consider a negative correlation, so that the
equilibrium trajectories must satisfy

$\displaystyle\xi_{i}$ $\displaystyle=J_{+}\sin({\gamma^{+}})t+\xi_{0},$
$\displaystyle\eta_{i}$ $\displaystyle\ =J_{-}\sin({\gamma^{+}})t+\frac{4\pi
i}{N}+\eta_{0}.$

Substituting in Eqs. (4) we find $v_{x}+v_{\theta}=J_{+}\sin\gamma^{+}$ and
$v_{x}-v_{\theta}=J_{-}\sin\gamma^{+}$. To study the stability of this
solution, we again add perturbations $\delta\xi_{i}$ and $\delta\eta_{i}$ to
the equilibrium and find their dynamics. We obtain

$\displaystyle\begin{split}\delta\dot{\xi_{i}}&=J^{+}\cos(\gamma^{+})\sum_{j}(\delta\xi_{j}-\delta\xi_{i})+J^{-}\sum_{j}\delta\eta_{j}\cos\left(\frac{4\pi}{N}(j-i)+\gamma^{-}\right),\\\
\delta\dot{\eta_{i}}&=J^{-}\cos(\gamma^{+})\sum_{j}(\delta\xi_{j}-\delta\xi_{i})+J^{+}\sum_{j}\delta\eta_{j}\cos\left(\frac{4\pi}{N}(j-i)+\gamma^{-}\right).\end{split}$
(10)

The dynamics of the perturbations can again be arranged using a $2N\times 2N$
block matrix, and the stability of the system analyzed by its eigenvalues. A
detailed derivation of the eigenvalues is shown in Appendix A. We obtain

$\displaystyle\begin{split}\lambda^{nPW}_{0}&=0,\\\
\lambda^{nPW}_{1}&=-J_{+}\cos(\gamma^{+}),\\\
\lambda^{nPW}_{2\pm}&=\frac{J_{+}}{2}\left(\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}-\cos(\gamma^{+})\right)\pm\frac{1}{2}\left[{J_{+}}^{2}\left(\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}+\cos(\gamma^{+})\right)^{2}-2{J_{-}}^{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}\cos(\gamma^{+})\right]^{1/2},\\\
\lambda^{nPW}_{(N-2)\pm}&=\frac{J_{+}}{2}\left(\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}-\cos(\gamma^{+})\right)\pm\frac{1}{2}\left[{J_{+}}^{2}\left(\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}+\cos(\gamma^{+})\right)^{2}-2{J_{-}}^{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}\cos(\gamma^{+})\right]^{1/2},\end{split}$ (11)

where $\lambda_{0}^{nPW}$ and $\lambda_{1}^{nPW}$ have multiplicities of $2$
and $(N-6)$, respectively. The superscript $PW$ stands for Phase Wave.
Considering a positive position-phase correlation leads to slightly different
eigenvalues. These differences, however, generate just a $\pi/2$ rotation of
the stability regions (as will be shown in the next section). In Appendix A,
we summarize the derivation of the eigenvalues for the positively correlated
Phase Wave states.

For $J=K$ the non-zero eigenvalues are

$\displaystyle\begin{split}\lambda_{1}^{nPW(J=K)}&=-J_{+}\cos(\gamma^{+}),\\\
\lambda_{2+}^{nPW(J=K)}&=\frac{J_{+}}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}},\\\ \lambda_{2,(N-2)-}^{nPW(J=K)}&=-J_{+}\cos(\gamma^{+}),\\\
\lambda_{(N-2)+}^{nPW(J=K)}&=\frac{J_{+}}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}},\\\ \end{split}$ (12)

and the regions where $\operatorname{Re}\\{\lambda^{nPW(J=K)}\\}$ are
negative, for $J_{+}>0$ are $\gamma^{+}\in[-\pi/2,\pi/2]$ and
$\gamma^{-}\in[\pi/2,3\pi/2]$, and for $J_{+}<0$ are
$\gamma^{+}\in[\pi/2,3\pi/2]$ and $\gamma^{-}\in[-\pi/2,\pi/2]$. The regions
for $J\neq K$, and for the positively correlated Phase Wave states are shown
in the next section. Moreover, the symmetry $1/K\rightarrow K$ still applies
for $J=1$, and so does the scaling in Eq. (9). The stability regions for $K$
and $1/K$ are, therefore, also the same for the Phase Wave states.

###  III.3 Asynchronous states

These states, such as the one shown in Figure 1c, are characterized by a
uniform and uncorrelated distribution of particles in position and phase. To
study their stability we take the limit of infinitely many oscillators and
assume a continuum of particles described by the density function
$\rho(x,\theta,t)\mathrm{d}x\mathrm{d}\theta$, which gives the fraction of
particles lying between $x+\mathrm{d}x$ and $\theta+\mathrm{d}\theta$ at time
$t$ [26]. The normalization condition for the density is

$\int_{0}^{2\pi}\int_{0}^{2\pi}\rho(x,\theta,t)\mathrm{d}x\mathrm{d}\theta=1,$
(13)

which allows us to rewrite the order parameters, introduced in Eqs. (5), as

$S_{\pm}e^{\mathrm{i}\mkern
1.0mu\phi_{\pm}}=\int_{0}^{2\pi}\int_{0}^{2\pi}e^{\mathrm{i}\mkern
1.0mu(x\pm\theta)}\rho(x,\theta,t)\mathrm{d}x\mathrm{d}\theta.$ (14)

Given that the description of this state is more intuitive in terms of $x$ and
$\theta$, we use Eqs. (3) for its analysis. To reduce the size of the
equations, however, we keep using the parameters $\xi$ and $\eta$ as defined
before. Then, the equations of motion give the velocity vector field governing
the behavior of the system:

$\displaystyle\begin{split}\dot{x}&=\frac{J}{2}S_{+}\sin(\phi_{+}-\xi+\gamma^{+})+\frac{J}{2}S_{-}\sin(\phi_{-}-\eta+\gamma^{-})\\\
\dot{\theta}&=\frac{K}{2}S_{+}\sin(\phi_{+}-\xi+\gamma^{+})-\frac{K}{2}S_{-}\sin(\phi_{-}-\eta+\gamma^{-}).\end{split}$
(15)

The temporal evolution of the density $\rho(x,\theta,t)$ is described by the
continuity equation

$\frac{\partial\rho}{\partial t}+\nabla(\rho\vec{v})=0,$ (16)

where $\vec{v}=(\dot{x},\dot{\theta})$ as in Eqs. (15). The uniform density
$\rho_{0}=\pi^{-2}/4$ with $S_{+}=S_{-}=0$ is an equilibrium incoherent state
and its stability can be studied by perturbing it as
$\rho=\rho_{0}+\delta\rho$ and analyzing the perturbation dynamics. The
perturbation analysis is detailed in Appendix C, and it leads to the
derivation of the eigenvalues

$\displaystyle\begin{split}\lambda^{oAS}_{1\pm}&=\frac{J_{+}}{8\pi^{2}}e^{\pm\mathrm{i}\mkern
1.0mu\gamma^{+}},\\\
\lambda^{oAS}_{2\pm}&=\frac{J_{+}}{8\pi^{2}}e^{\pm\mathrm{i}\mkern
1.0mu\gamma^{-}}.\end{split}$ (17)

The Static Asynchronous state will be stable when the real part of these
eigenvalues are negative, that is, for $J_{+}>0$ when
$\gamma^{\pm}\in[\pi/2,3\pi/2]$, and for $J_{+}<0$ when
$\gamma^{\pm}\in[-\pi/2,\pi/2]$.

##  IV Stability diagrams

The analytical results obtained in the previous section can be summarized with
graphic representations of the Sakaguchi Swarmalators’ stability regions
spanned for $\gamma^{\pm}\in[-\pi$, $\pi]$. We fix $J=1$ in all diagrams, as
this specific setup allows us to observe the stability symmetry expected when
using $K$ and $1/K$ (Figures 4b, 4c, 4e, and 4f).

The simplest case, where $J=K$, is shown in Figure 4a. As calculated in Eqs.
(8), (12), and (17), the regions where each state emerges are just
complementing squares. However, for $K>0$ and different from $J$ (Figures 4b
and 4c), the Phase Wave squared regions deform giving rise to the formation of
stability regions that intersect. These are regions of multi-stability, where
the system converges to either the Synchronous or Phase Wave states, depending
on the initial conditions. Once $K$ becomes negative, the Synchronous square
regions, depicted for $J\geq K$, split into four triangles, Figs. 4e and 4f,
that split again for $K<-J$, Figs. 4g, and 4h. Even more interesting is the
behavior of the Phase Wave regions, which for $K$ approaching $-J$ from the
right form intersecting $\pi/2$-rotated stripes (Figure 4e) that, when $K$
increases towards $0$, become fully intersected circles (Figure 4f).
Similarly, the split of the Synchronous square and the Phase Wave stripes,
which become circles, are found when $K$ is set below $-J$ (Figures 4g and
4h). Despite the similarities, however, these stability regions are
$\pi$-translated from the $-J<K<0$ regions described before, in both the
$\gamma^{-}$ and $\gamma^{+}$ axes. We note that no such multi-stable regions
exist in the Kuramoto-Sakaguchi model, suggesting that they depend on the
interplay between the two degrees of freedom $\theta$ and $x$.

A special case takes place when $K=-J$, which corresponds to $J_{+}=0$, and
leads to the vanishing of negative eigenvalues in all states. Under these
circumstances, the system exhibits the Active Asynchronous state, shown in
Figure 3b. It is worth noting that, for these parameters, Eqs. (2) reduce to a
Hamiltonian system with
$H=(J/2N)\sum_{i,j}\sin(x_{j}-x_{i})\sin(\theta_{j}-\theta_{i})$. However, for
frustration parameters $(\alpha,\beta)\neq 0$, the system loses the
Hamiltonian structure and, therefore, a constant of motion. The stability
regions when $K=0$ are shown in Figure 4d, and, despite these appearing to be
well-defined, their nature is very susceptible to small numerical changes. The
Noisy Phase Wave and Turbulent states (introduced in Figures 3a and 3c) are
also found in the stability diagrams, as pointed by red markers in Figures 4e,
4f, 4g, and 4h. These, given their disordered nature, do not belong to any of
the stable state’s regions but are positioned in the blank spaces. Finally,
the intriguing ring state, introduced in Figure 2c, turns out to be an
intermediate state when entering the Synchronous states’ stable regions (as
depicted by the blue star in Figure 4g), which gives meaning to its clustered
behavior. This state allows us to remark that, at the boundaries, linear
stability analysis is not enough to decide the nature of the equilibrium.

Figure 4: Stability regions computed using the eigenvalues obtained from the
perturbation analyses. All figures were obtained for $J=1$ and $N=500$. In b),
the diagram shows the regions for both $K=1/5$ and $K=5$. Similarly, in c),
the diagram corresponds to both $K=1/500$ and $K=500$. In e) and f), $K=-1/5$
and $K=-1/500$, respectively. And, in g) and h), $K=-5$ and $K=-500$,
respectively. Red circles, stars and squares are positioned in regions where
states Turbulent, Noisy Active Phase Wave, and a combination of these two
emerge, respectively. The blue star in g) corresponds to one instance of Ring
state.

In Figure 5, we show heatmaps that complement the stability diagrams,
presented in Figure 4, for the case where $J=K$. Each diagram is obtained for
a system of $N=500$ particles at its state after $10^{4}$ time steps.
Intensities are positioned according to specific values of $\gamma^{\pm}$ used
for the computation. In Figures 5a and 5b, these correspond to $S_{+}$ and
$S_{-}$, respectively, calculated using Eqs. (5). For Figure 5c, we use the
additional parameter $S_{v}=1/N\left|\sum_{j}{\dot{x}_{j}}\right|$, which is
an indicator of the average velocity in the system.

Two main observations can be made by contrasting Figures 4a and 5. Firstly, as
shown in the respective $S_{\pm}$ heatmaps, we can identify regions where the
system converges to positively or negatively correlated states, which may
represent either the Synchronous or Phase Wave states. Secondly, although
convergence to any stable state can be proven by the eigenvalues, the
definitions of $\xi$ and $\eta$ used for the stability analyses do not
guarantee whether the system is static or active, even when analyzing
$S_{\pm}$. However, the use of $S_{v}$ provides additional information that
allows us to overcome these limitations. As shown in Figure 5c, clear
partitions consistent with the regions in Figure 4a are displayed, and
intensities reveal the velocity dependence on $\gamma^{\pm}$, which is
coherent with the equilibrium trajectories defined in Section III for the
Synchronous and Phase Wave states.

Figure 5: Order parameter heatmaps for $J=K=1$ and $N=500$, after $10^{4}$
time steps. Each intensity corresponds to the value of the respective order
parameter for specific $(\gamma^{+},\gamma^{-})$ values. The diagrams show a)
positive and b) negative position-phase correlation regions, and c) the
average velocities.

##  V Conclusions

We studied the effects of frustration induced disorder on a 1D Swarmalators
system. Motivated by the work of Sakaguchi [24], we modified the original
system introduced in [20], by including frustration parameters intended to
break the coherence of the system in both position and phase spaces.

The most striking feature of the model is the emergence of active states for
non-zero frustration parameters. These states remind us of the ones found in
the 2D Swarmalators model [12], that still lack a complete analytical
explanation. In our model, however, we were able to find analytical solutions
for the stability regions of each ordered state, independent of its static or
active nature. Additionally, numerical computations allowed us to find regions
where disordered active states emerge. In these states, despite the incoherent
behavior exhibited by the particles, clear position-phase patterns can still
be observed, which suggests that their analytical study could also be
performed using different tools.

From the stability analyses we see that, in contrast to the original 1D
Swarmalators model, the frustration parameters provide us flexibility to find
ordered states for any fixed $(J,K)$ values. That is, for a specific $(J,K)$
setup, we can find Synchronous, positively or negatively correlated Phase
Wave, or Asynchronous states just by tuning the values of $\gamma^{\pm}$. The
disordered states, however, have been spotted only for $K<0$. Even more exotic
is the emergence of the Active Asynchronous case, which shows up only when
$J=-K$.

Although Eqs. (3) have been defined in terms of an internal phase $\theta$ and
a spatial coordinate $x$, we can think of the phase variable as another
periodic spatial coordinate $y$, so that the scatter plots in Figures 1, 2,
and 3 could represent particles’ positions in the periodic Cartesian plane (a
torus). The Active Asynchronous and Turbulent states are then similar to the
patterns displayed by chiral rollers in [8] or ram semen in [27]. The
turbulent state, in particular, is of specific interest for future studies, as
it shows the emergence of vortices and eddy-like structures (see movies S6 and
S7 in Supplemental Material) even for finite number of particles. Moreover,
under this Cartesian setup, our results can also be extended to swarming-only
systems whose position degrees of freedom interact.

An interesting take on the Sakaguchi Swarmalators model would be to consider
asymmetrical frustrations (i.e. parameters affecting only the sines or
cosines), or even considering distributed couplings and frustrations, as done
in [28] for the Kuramoto model. In general, since the Kuramoto-Sakaguchi model
and the concept of frustration have been widely studied [29, 30, 31, 32], the
study of Sakaguchi Swarmalators can be expanded following these ideas.

###### Acknowledgements.

It is a pleasure to thank Kevin P. O’Keeffe for helpful comments and
suggestions. This work was partly supported by FAPESP grant 2021/14335-0 and
CNPq grant 301082/2019‐7 (MAMA) and FAPESP grant 2021/04251-4 (JUFL).

##  Appendix A Eigenvalues for coherent states

The stability calculations, described in subsections III.1 and III.2, show
that perturbation dynamics can be arranged as

$\dot{\vec{\delta_{*}}}=\mathbf{R}\vec{\delta_{*}},$ (18)

where, for each system’s state, the vector $\vec{\delta_{*}}$ is composed of
the individual perturbations $\delta\xi_{i}$ and $\delta\eta_{i}$, and

$\mathbf{R}=\begin{bmatrix}\mathbf{R_{11}}&\mathbf{R_{12}}\\\
\mathbf{R_{21}}&\mathbf{R_{22}}\end{bmatrix},$

is a matrix of circulant blocks. The structure of $\mathbf{R}$ allows us to
find its eigenvalues $\lambda$ for all states, following the general procedure
described below.

The eigenvalues are solutions of the equation

$\det(\mathbf{R}-\lambda\mathbf{I_{2N}})=0,$

where $\mathbf{I_{2N}}$ is the identity matrix of dimension $2N$. However,
since $[\mathbf{R_{11}},\mathbf{R_{21}}]=0$ holds for both the Synchronous and
Phase Wave cases, we rewrite the equation for the determinant as

$\det(\mathbf{R}-\lambda\mathbf{I_{2N}})=\det(\mathbf{M}),$

where
$\mathbf{M}=(\mathbf{R_{11}-\lambda\mathbf{I_{N}}})(\mathbf{R_{22}-\lambda\mathbf{I_{N}}})-\mathbf{R_{21}}\mathbf{R_{12}}$
is also circulant. Then, the determinant of $\mathbf{M}$ can be computed using
the general solution for circulant matrices

$\det(\mathbf{M})=\prod_{k=0}^{N-1}\left(M_{11}+M_{12}\zeta^{k}+\cdots+M_{1N}\zeta^{(N-1)k}\right),$
(19)

where $\zeta$ is a primitive $N$-th root of unity, and the eigenvalues
$\lambda$ are found by equating the resulting equations inside the parenthesis
to zero.

In the next subsections, we describe the solution to the eigenvalue problems
for the Synchronous and Phase Wave states using the procedure described above.
In each of these states, the blocks composing $\mathbf{R}$ have different
structures that, however, can be considered special instances of circulant
matrices. To differentiate between Synchronous and Phase Wave states, we use
respectively the superscripts SS and PW on matrices and their elements. Since
the Phase Wave state has two instances, we add the letters $n$ and $p$ in
front of the superscripts to differentiate according to the negative or
positive nature of the position-phase correlation. Additionally, to reduce the
size of some expressions, we use $s_{*}$ and $c_{*}$ to represent $\sin(*)$
and $\cos(*)$ functions, respectively.

###  A.1 Synchronous states

Arranging Eqs. (6) as Eq. (18) leads to a matrix $\mathbf{R}^{SS}$ composed by
blocks

$\displaystyle\mathbf{R}^{SS}_{\mathbf{{11}}}$
$\displaystyle=J^{+}\cos(\gamma^{+})\mathbf{R}_{\dagger},$
$\displaystyle\mathbf{R}^{SS}_{\mathbf{{12}}}$
$\displaystyle=J^{-}\cos(\gamma^{-})\mathbf{R}_{\dagger},$
$\displaystyle\mathbf{R}^{SS}_{\mathbf{{21}}}$
$\displaystyle=J^{-}\cos(\gamma^{+})\mathbf{R}_{\dagger},$
$\displaystyle\mathbf{R}^{SS}_{\mathbf{{22}}}$
$\displaystyle=J^{+}\cos(\gamma^{-})\mathbf{R}_{\dagger},$

where

$\mathbf{R}_{\dagger}=\begin{pmatrix}1-N&1&\cdots&1\\\ 1&1-N&\cdots&1\\\
\vdots&\vdots&\ddots&\vdots\\\ 1&1&\cdots&1-N\end{pmatrix}.$

The composition of the block matrix $\mathbf{R}^{SS}$ allows us to infer that
the off-diagonal terms of $\mathbf{M}^{SS}$ will all be the same. Thus, the
only relevant elements to calculate $\det(\mathbf{M})$ are

$\displaystyle M_{11}^{SS}$ $\displaystyle=\lambda^{2}-\lambda
J^{+}(1-N)(c_{\gamma^{-}}+c_{\gamma^{+}})\,+$ $\displaystyle\quad
N(1-N)c_{\gamma^{+}}c_{\gamma^{-}}({J^{-}}^{2}-{J^{+}}^{2}),$ $\displaystyle
M_{12}^{SS}$ $\displaystyle=-\lambda
J^{+}(c_{\gamma^{-}}+c_{\gamma+})+Nc_{\gamma^{+}}c_{\gamma^{-}}({J^{-}}^{2}-{J^{+}}^{2}),$

and Eq. (19) can be rewritten as

$\det(\mathbf{M}^{SS})=\prod_{k=0}^{N-1}\left(M_{11}^{SS}+M_{12}^{SS}\sum_{r=1}^{N-1}\zeta^{rk}\right),$
(20)

where

$\sum_{r=1}^{N-1}\zeta^{rk}=-1$

for $k\neq 0$.

Solving Eq. (20) leads to the product of $N$ quadratic equations, where $N-1$
of them are replicas. Equating this product to zero allows us to get the
eigenvalue expressions shown in Eqs. (7).

###  A.2 Negatively correlated Phase Wave states

For these states, obtain the blocks

$\displaystyle\mathbf{R}^{nPW}_{\mathbf{{11}}}$
$\displaystyle=J^{+}\cos(\gamma^{+})\mathbf{R}_{\dagger},$
$\displaystyle\mathbf{R}^{nPW}_{\mathbf{{12}}}$
$\displaystyle=J^{-}\mathbf{R}^{-}_{*},$
$\displaystyle\mathbf{R}^{nPW}_{\mathbf{{21}}}$
$\displaystyle=J^{-}\cos(\gamma^{+})\mathbf{R}_{\dagger},$
$\displaystyle\mathbf{R}^{nPW}_{\mathbf{{22}}}$
$\displaystyle=J^{+}\mathbf{R}^{-}_{*},$

where

$\mathbf{R}^{\pm}_{*}=\begin{pmatrix}\cos(\gamma^{\pm})&\cos\left(4\pi\frac{1}{N}+\gamma^{\pm}\right)&\cdots&\cos\left(4\pi\frac{N-1}{N}+\gamma^{\pm}\right)\\\
\cos\left(4\pi\frac{N-1}{N}+\gamma^{\pm}\right)&\cos(\gamma^{\pm})&\cdots&\cos\left(4\pi\frac{N-2}{N}+\gamma^{\pm}\right)\\\
\vdots&\vdots&\ddots&\vdots\\\
\cos\left(4\pi\frac{1}{N}+\gamma^{\pm}\right)&\cos\left(4\pi\frac{2}{N}+\gamma^{\pm}\right)&\cdots&cos(\gamma^{\pm})\end{pmatrix}.$

Although we use only $\mathbf{R}^{+}_{*}$ to describe the $\mathbf{R}^{nPW}$
blocks, the matrix $\mathbf{R}^{-}_{*}$ will be used in the next subsection
when describing the $\mathbf{R}^{pPW}$ blocks corresponding to the positive
correlated Phase Wave states.

The blocks in $\mathbf{R}^{nPW}$ have different off-diagonal elements, which
make the structure of $\mathbf{M}^{nPW}$ less intuitive. In this case the
elementz needed to compute the determinant are

$\displaystyle\begin{split}M_{11}^{nPW}&=\lambda^{2}-\lambda
J^{+}\left[c_{\gamma^{-}}+(1-N)c_{\gamma^{+}}\right]+N\left({J^{-}}^{2}-{J^{+}}^{2}\right)c_{\gamma^{+}}c_{\gamma^{-}},\\\
M_{1(r+1)}^{nPW}&=-\lambda
J^{+}\left[\cos\left(\frac{4\pi}{N}r+\gamma^{-}\right)+c_{\gamma^{+}}\right]+N\left({J^{-}}^{2}-{J^{+}}^{2}\right)c_{\gamma^{+}}\cos\left(\frac{4\pi}{N}r+\gamma^{-}\right),\end{split}$

and Eq. (19) can be rewritten as

$\det(\mathbf{M}^{nPWS})=\prod_{k=0}^{N-1}\left(M_{11}^{nPW}+\sum_{r=1}^{N-1}M_{1(r+1)}^{nPW}\zeta^{rk}\right).$
(21)

Setting Eq. (21) to zero should return the eigenvalues shown in Eqs.(11).
However, since elements $M_{1(r+1)}^{nPW}$ are all different, this solution is
a bit more intricate. In Appendix B, we show the simplification of Eq. (21)
that allows us to get analytical solutions for the eigenvalues.

###  A.3 Positively correlated Phase Wave states

We start by summarizing the perturbation analysis since this was removed from
the main text for the sake of clarity. For the Phase Wave states that exhibit
a positive position-phase correlation, the equilibrium trajectories are

$\displaystyle\xi_{i}$ $\displaystyle=J_{-}\sin({\gamma^{-}})t+\frac{4\pi
i}{N}+\xi_{0},$ $\displaystyle\eta_{i}$
$\displaystyle=J_{+}\sin({\gamma^{-}})t+\eta_{0}.$

Adding perturbations to the equilibrium solutions we obtain

$\displaystyle\begin{split}\delta\dot{\xi_{i}}&=J^{-}c_{\gamma^{-}}\sum_{j}(\delta\eta_{j}-\delta\eta_{i})+J^{+}\sum_{j}\delta\xi_{j}C_{ij+},\\\
\delta\dot{\eta_{i}}&=J^{+}c_{\gamma^{-}}\sum_{j}(\delta\eta_{j}-\delta\eta_{i})+J^{-}\sum_{j}\delta\xi_{j}C_{ij+},\end{split}$

where $C_{ij+}=\cos\left(\frac{4\pi}{N}(j-i)+\gamma^{+}\right)$. This can be
arranged in blocks

$\displaystyle\mathbf{R}^{pPW}_{\mathbf{{11}}}$
$\displaystyle=J^{+}\mathbf{R}^{+}_{*},$
$\displaystyle\mathbf{R}^{pPW}_{\mathbf{{12}}}$
$\displaystyle=J^{-}\cos(\gamma^{-})\mathbf{R}_{\dagger},$
$\displaystyle\mathbf{R}^{pPW}_{\mathbf{{21}}}$
$\displaystyle=J^{-}\mathbf{R}^{+}_{*},$
$\displaystyle\mathbf{R}^{pPW}_{\mathbf{{22}}}$
$\displaystyle=J^{+}\cos(\gamma^{-})\mathbf{R}_{\dagger}.$

Despite the differences between $\mathbf{R}^{nPW}$ and $\mathbf{R}^{pPW}$,
elements of $\mathbf{M}^{nPW}$ and $\mathbf{M}^{pPW}$ differ only by a swap of
$\gamma^{+}$ and $\gamma^{-}$. Thus, we find

$\displaystyle\begin{split}M_{11}^{pPW}&=\lambda^{2}-\lambda
J^{+}\left[(1-N)c_{\gamma^{-}}+c_{\gamma^{+}}\right]+N\left({J^{-}}^{2}-{J^{+}}^{2}\right)c_{\gamma^{+}}c_{\gamma^{-}},\\\
M_{1(r+1)}^{pPW}&=-\lambda
J^{+}\left[c_{\gamma^{-}}+\cos\left(\frac{4\pi}{N}r+\gamma^{+}\right)\right]+N\left({J^{-}}^{2}-{J^{+}}^{2}\right)\cos\left(\frac{4\pi}{N}r+\gamma^{+}\right)c_{\gamma^{-}}.\end{split}$

Then, $\det(\mathbf{M}^{pPW})$ has the same form of Eq. (21) and it can be
solved using the simplification shown in Appendix B. The solution leads to the
eigenvalues

$\displaystyle\begin{split}\lambda^{pPW}_{0}&=0,\\\
\lambda^{pPW}_{1}&=-J_{+}\cos(\gamma^{-}),\\\
\lambda^{pPW}_{2\pm}&=\frac{J_{+}}{2}\left(\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{+}}-\cos(\gamma^{-})\right)\pm\frac{1}{2}\left[{J_{+}}^{2}\left(\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{+}}+\cos(\gamma^{-})\right)^{2}-2{J_{-}}^{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{+}}\cos(\gamma^{-})\right]^{1/2},\\\
\lambda^{pPW}_{(N-2)\pm}&=\frac{J_{+}}{2}\left(\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{+}}-\cos(\gamma^{-})\right)\pm\frac{1}{2}\left[{J_{+}}^{2}\left(\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{+}}+\cos(\gamma^{-})\right)^{2}-2{J_{-}}^{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{+}}\cos(\gamma^{-})\right]^{1/2}.\end{split}$ (22)

##  Appendix B Simplification

We are interested in finding the solutions of a quadratic equation of the form

$\lambda^{2}+\lambda\phi+\omega=0,$

where, from Eq. (21),

$\displaystyle\begin{split}\phi&=-J^{+}\left[c_{\gamma^{-}}+c_{\gamma^{+}}\left(1-N+\sum_{r=1}^{N-1}\zeta^{rk}\right)+\sum_{r=1}^{N-1}\zeta^{rk}\cos\left(\frac{4\pi}{N}r+\gamma^{-}\right)\right],\\\
\omega&=N({J^{-}}^{2}-{J^{+}}^{2})c_{\gamma^{+}}\left[c_{\gamma^{-}}+\sum_{r=1}^{N-1}\zeta^{rk}\cos\left(\frac{4\pi}{N}r+\gamma^{-}\right)\right].\end{split}$

We expand the sum in the right hand side of these equations as

$\displaystyle\sum_{r=1}^{N-1}\zeta^{rk}$
$\displaystyle\cos\left(\frac{4\pi}{N}r+\gamma^{-}\right)=\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}\sum_{r=1}^{N-1}e^{\mathrm{i}\mkern 1.0mur(4\pi+2\pi k)/N}$
$\displaystyle+\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}\sum_{r=1}^{N-1}e^{-\mathrm{i}\mkern 1.0mur(4\pi-2\pi k)/N}.$

Then, for $k\neq 2$,

$\displaystyle\begin{split}\phi&=\begin{cases}0,&\text{if}\ k=0\\\
J_{+}c_{\gamma^{+}},&\text{otherwise}\end{cases},\\\ \omega&=0,\end{split}$

which lead to pairs of eigenvalues where one of them is real and the other one
is zero, or both are zero ($k=0$). For $k=2$,

$\displaystyle\begin{split}\phi&=-J_{+}\left(\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}-c_{\gamma^{+}}\right),\\\
\omega&=\frac{1}{2}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}\left({J_{-}}^{2}-{J_{+}}^{2}\right)c_{\gamma^{+}},\end{split}$

and $k=(N-2)$,

$\displaystyle\begin{split}\phi&=-J_{+}\left(\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}-c_{\gamma^{+}}\right),\\\
\omega&=\frac{1}{2}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}\left({J_{-}}^{2}-{J_{+}}^{2}\right)c_{\gamma^{+}},\end{split}$

which lead to pairs of complex conjugate eigenvalues.

##  Appendix C Perturbation analysis for incoherent states

Incoherent states are characterized by $S_{\pm}=0$ and, therefore,
$\vec{v}=(\dot{x},\dot{\theta})=\vec{0}$. Thus, the homogeneous density
$\rho_{0}=\pi^{-2}/4$ is a static solution of the continuity equation (14).

Adding a small perturbation to the equilibrium state,
$\rho=\rho_{0}+\delta\rho$, and using Eq. (16), we find that the temporal
evolution of the perturbed state is governed by

$\frac{\partial}{\partial{t}}{\delta\rho}=-\nabla\left(\delta\rho\right)\vec{v},$
(23)

where, from Eq. (13)

$\int_{0}^{2\pi}\int_{0}^{2\pi}\delta\rho(x,\theta,t)\mathrm{d}x\mathrm{d}\theta=0.$
(24)

To first order in $\delta\rho(x,\theta,t)$, we find that

$S_{\pm}^{1}e^{\mathrm{i}\mkern
1.0mu\phi_{\pm}}=\int_{0}^{2\pi}\int_{0}^{2\pi}e^{\mathrm{i}\mkern
1.0mu(x\pm\theta)}\delta\rho(x,\theta,t)\mathrm{d}x\mathrm{d}\theta,$ (25)

which leads to

$\displaystyle\begin{split}\frac{\partial}{\partial
t}\delta\rho&=\frac{J_{+}}{4\pi^{2}}\left(S^{1}_{+}\cos(\phi_{+}-\xi+\gamma^{+})\right.\\\
&\left.+\,S^{1}_{-}\cos(\phi_{-}-\eta+\gamma^{-})\right).\end{split}$ (26)

Expanding $\delta\rho$ in Fourier series

$\delta\rho=\sum_{m,n}f_{m,n}(t)e^{\mathrm{i}\mkern 1.0mu(mx+n\theta)},$ (27)

and comparing with Eq. (26) we see that the only relevant terms are $f_{\pm
1,\pm 1}$. We obtain

$\displaystyle\begin{split}\dot{f}_{1,1}(t)&=\frac{J_{+}}{8\pi^{2}}e^{-\mathrm{i}\mkern
1.0mu\gamma^{-}}f_{1,1}(t),\\\
\dot{f}_{-1,1}(t)&=\frac{J_{+}}{8\pi^{2}}e^{\mathrm{i}\mkern
1.0mu\gamma^{-}}f_{-1,1}(t),\\\
\dot{f}_{1,-1}(t)&=\frac{J_{+}}{8\pi^{2}}e^{-\mathrm{i}\mkern
1.0mu\gamma^{+}}f_{1,-1}(t),\\\
\dot{f}_{-1,-1}(t)&=\frac{J_{+}}{8\pi^{2}}e^{\mathrm{i}\mkern
1.0mu\gamma^{+}}f_{-1,-1}(t).\end{split}$ (28)

Finally, writting $f(t)=\bar{f}e^{t\lambda}$, we can solve Eqs. (28) to get
the eigenvalues shown in Eqs. (17).

## References

  * Acebrón  _et al._ [2005] J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, Reviews of modern physics 77, 137 (2005).
  * Rodrigues  _et al._ [2016] F. A. Rodrigues, T. K. D. M. Peron, P. Ji, and J. Kurths, Physics Reports 610, 1 (2016), arXiv:1511.07139 .
  * Matthews  _et al._ [1991] P. C. Matthews, R. E. Mirollo, and S. H. Strogatz, Physica D: Nonlinear Phenomena 52, 293 (1991).
  * Aoyagi [1995] T. Aoyagi, Physical review letters 74, 4075 (1995).
  * Katz  _et al._ [2011] Y. Katz, K. Tunstrøm, C. C. Ioannou, C. Huepe, and I. D. Couzin, Proceedings of the National Academy of Sciences 108, 18720 (2011).
  * Cavagna  _et al._ [2023] A. Cavagna, L. Di Carlo, I. Giardina, T. S. Grigera, S. Melillo, L. Parisi, G. Pisegna, and M. Scandolo, Nature Physics , 1 (2023).
  * Aihara  _et al._ [2014] I. Aihara, T. Mizumoto, T. Otsuka, H. Awano, K. Nagira, H. G. Okuno, and K. Aihara, Scientific reports 4, 3891 (2014).
  * Zhang  _et al._ [2020] B. Zhang, A. Sokolov, and A. Snezhko, Nature communications 11, 4401 (2020).
  * Giomi  _et al._ [2013] L. Giomi, N. Hawley-Weld, and L. Mahadevan, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, 20120637 (2013).
  * Tan  _et al._ [2022] T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Nature 607, 287 (2022).
  * Creppy  _et al._ [2016] A. Creppy, F. Plouraboué, O. Praud, X. Druart, S. Cazin, H. Yu, and P. Degond, Journal of The Royal Society Interface 13, 20160575 (2016).
  * O’Keeffe  _et al._ [2017] K. P. O’Keeffe, H. Hong, and S. H. Strogatz, Nature communications 8, 1 (2017).
  * Lizarraga and de Aguiar [2020] J. U. Lizarraga and M. A. de Aguiar, Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 053112 (2020).
  * Hong [2018] H. Hong, Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 103112 (2018).
  * Sar  _et al._ [2022] G. K. Sar, S. N. Chowdhury, M. Perc, and D. Ghosh, New Journal of Physics 24, 043004 (2022).
  * Lee  _et al._ [2021] H. K. Lee, K. Yeo, and H. Hong, Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 033134 (2021).
  * Jiménez-Morales [2020] F. Jiménez-Morales, Physical Review E 101, 062202 (2020).
  * Japón  _et al._ [2022] P. Japón, F. Jiménez-Morales, and F. Casares, Cells & Development 169, 203726 (2022).
  * Hong  _et al._ [2023] H. Hong, K. P. O’Keeffe, J. S. Lee, and H. Park, arXiv preprint arXiv:2302.11820 (2023).
  * O’Keeffe  _et al._ [2022] K. O’Keeffe, S. Ceron, and K. Petersen, Physical Review E 105, 014211 (2022).
  * O’Keeffe and Hong [2022] K. O’Keeffe and H. Hong, Physical Review E 105, 064208 (2022).
  * Sar  _et al._ [2023] G. K. Sar, D. Ghosh, and K. O’Keeffe, Physical Review E 107, 024215 (2023).
  * Yoon  _et al._ [2022] S. Yoon, K. O’Keeffe, J. Mendes, and A. Goltsev, Physical Review Letters 129, 208002 (2022).
  * Sakaguchi and Kuramoto [1986] H. Sakaguchi and Y. Kuramoto, Progress of Theoretical Physics 76, 576 (1986).
  * Ceron  _et al._ [2023] S. Ceron, K. O’Keeffe, and K. Petersen, Nature Communications 14, 940 (2023).
  * Strogatz and Mirollo [1991] S. H. Strogatz and R. E. Mirollo, Journal of Statistical Physics 63, 613 (1991).
  * Schoeller  _et al._ [2020] S. F. Schoeller, W. V. Holt, and E. E. Keaveny, Philosophical Transactions of the Royal Society B 375, 20190384 (2020).
  * Daido [1992] H. Daido, Physical review letters 68, 1073 (1992).
  * Omel’Chenko and Wolfrum [2012] E. Omel’Chenko and M. Wolfrum, Physical review letters 109, 164101 (2012).
  * Manoranjani  _et al._ [2021] M. Manoranjani, S. Gupta, and V. Chandrasekar, Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 083130 (2021).
  * Arnaudon  _et al._ [2022] A. Arnaudon, R. L. Peach, G. Petri, and P. Expert, Communications Physics 5, 211 (2022).
  * de Aguiar [2023] M. A. M. de Aguiar, Phys. Rev. E 107, 044205 (2023).


# On p-groups of maximal class

Noureddine Snanou1

###### Abstract.

Recall that a p-group of order $p^{n}>p^{3}$ is of maximal class, if its
nilpotency class is $n-1$. In this paper, we recall some preliminaries of
nilpotent groups and give some basic material about p-groups of maximal class.
Furthermore, we introduce the fundamental subgroup of a p-group of maximal
class by specifying its importance in the general theory of p-groups of
maximal class.

Keywords: p-groups of maximal class, fundamental subgroup, nilpotency class.
Mathematics subject classification: 20D15.

##  1\. Introduction

A group of order $p^{n}>p^{3}$ and nilpotency class $n-1$, is said to be of
maximal class. These groups have been studied by various authors, the basic
material about these groups is to be found in Blackburn’s paper [5]. A more
modern reference for the theory of p-groups of maximal class is section $14$
of Chapter III in Huppert’s book [7]. The p-groups of maximal class with an
abelian maximal subgroup were completely classified by A. Wiman in [11]. The
most results about these groups that we present in this paper can be found in
[1, 2, 3]. These results play a fundamental role in finite p-group theory.
Throughout this paper, we use the standard notation, such as in [1].

The paper is organized as follows: In the second section, we recall some
preliminaries of nilpotent groups. The third section covers the basic material
about p-groups of maximal class. In section 4, we introduce a subgroup of a
p-group of maximal class called the fundamental subgroup. This group plays a
fundamental role in the development of the general theory of p-groups of
maximal class.

##  2\. Preliminaries

Let $G$ be a group and $H$ be a subgroup of $G$. An element $g\in G$
normalizes $H$ if $gHg^{-1}=H$. We call $N_{G}(H)=\\{g\in G|gHg^{-1}=H\\}$ the
normalizer of $H$ in $G$. An element $g\in G$ centralizes $H$ if $ghg^{-1}=h$
for any $h\in H$. We call $C_{G}(H)=\\{g\in G|ghg^{-1}=h,\forall h\in H\\}$
the centralizer of $H$ in $G$. If $H=G$, then $Z(G)=C_{G}(G)$ is called the
center of $G$.

Note that for a subgroup $H$ of $G$, $C_{G}(H)=G$ if and only if $H\leq Z(G)$.
It is easy to see that, for any subgroup $H$, $N_{G}(H)$ and $C_{G}(H)$ are
subgroups of G with $C_{G}(H)\leq N_{G}(H)$, and that if $H\leq G$ then $H\leq
N_{G}(H)$. The N/C-theorem [1, Introduction, Proposition 12] asserts that if
$H\leq G$, then the quotient group $N_{G}(H)/C_{G}(H)$ is isomorphic to a
subgroup of $Aut(H)$. In particular, $G/Z(G)$ is isomorphic to a subgroup of
$Aut(G)$.

Let $G$ be a group. Set $Z_{0}(G)=\\{1\\},Z_{1}(G)=Z(G)$. Suppose that
$Z_{i}(G)$ has been defined for $i\leq k$. Define $Z_{k+1}(G)$ as follows:
$Z_{k+1}(G)/Z_{k}(G)=Z(G/Z_{k}(G))$. The chain $\\{1\\}=Z_{0}(G)\leq
Z_{1}(G)\leq...\leq Z_{k}(G)\leq...$ is said to be the upper central series of
$G$. All members of that series are characteristic in $G$.

######  Definition 2.1.

For elements $x,y\in G$, their commutator $x^{-1}y^{-1}xy$ is written as
$[x,y]$. If $X,Y\subseteq G$, then $[X,Y]$ is the subgroup generated by all
commutators $[x,y]$ with $x\in X,y\in Y$.

The lower central series $G=K_{1}(G)\geq K_{1}(G)\geq K_{2}(G)\geq...$ of $G$
is defined as follows: $K_{1}(G)=G,K_{i+1}(G)=[K_{i}(G),G],i>0$. All members
of that series are characteristic in $G$. We have $K_{i}(G)/K_{i+1}(G)\leq
Z(G/K_{i}(G))$. If $H\leq G$, then $K_{i}(H)\leq K_{i}(G)$ for all $i$.

Since $[y,x]=[x,y]^{-1}$, we have $[Y,X]=[X,Y]$. We write $[G,G]=G^{\prime}$,
the subgroup $G^{\prime}$ is called the commutator (or derived subgroup) of
$G$. We also write $G^{0}=G$, $G^{\prime}=G^{1}$. Then the subgroup
$G^{i+1}=[G^{i},G^{i}]$ is called the (i+1)th derived subgroup of $G$, $i\geq
0$. The chain $G=G^{0}\geq G^{1}\geq...\geq G^{n}\geq...$ is called the
derived series of $G$. All members of this series are characteristic in $G$
and all factors $G^{i}/G^{i+1}$ are abelian. The group $G$ is said to be
solvable if $G^{n}={1}$ for some $n$.

######  Definition 2.2.

A group $G$ is said to be nilpotent if the upper central series of $G$
contains $G$. In other words, $G$ is nilpotent of class $c$, if $Z_{c}(G)=G$
but $Z_{c-1}(G)<G$, we write $c=cl(G)$. In particular, the class of the
identity group is $0$ and the class of a nonidentity abelian group is $1$.

######  Lemma 2.3.

The following are equivalent:

  1. (1)

$G$ is nilpotent of class $c$.

  2. (2)

$G=K_{1}(G)>K_{2}(G)>K_{3}(G)>...>K_{c+1}(G)=\langle 1\rangle$.

  3. (3)

$\langle 1\rangle=Z_{0}(G)<Z_{1}(G)<...<Z_{c-1}(G)<Z_{c}(G)=G$.

######  Theorem 2.4.

Let $G$ be a p-group of order $p^{m}\geq p^{2}$. Then:

  1. (1)

$G$ is nilpotent of class at most $m-1$.

  2. (2)

If $G$ has nilpotency class $c$ then $|G:Z_{c-1}(G)|\geq p^{2}$.

  3. (3)

The maximal subgroups of $G$ are normal and of index $p$.

  4. (4)

$|G:G^{\prime}|\geq p^{2}$.

######  Corollary 2.5.

Let $G$ be a p-group and let $N$ be a normal subgroup of $G$ of index
$p^{i}\geq p^{2}$. Then $K_{i}(G)\leq N$.

###### Proof.

The group $G/N$ has order $p^{i}\geq p^{2}$. It follows from part $(1)$ of the
Theorem 2.4 that $G/N$ has class $\leq i-1$ and consequently
$K_{i}(G/N)=\overline{1}$. Since $K_{i}(G/N)=K_{i}(G)N/N$, this proves that
$K_{i}(G)\leq N$. ∎

##  3\. P-group of maximal class

Recall that a group of order $p^{m}$ is of maximal class, if $cl(G)=m-1>1$. Of
course, any group of order $p^{2}$ and any nonabelian p-group of order $p^{3}$
have maximal class. If $G$ is a group of maximal class and of order $p^{m}$,
then the lower and upper central series of $G$ are:

$G=K_{1}(G)>K_{2}(G)>K_{3}(G)>...>K_{m}(G)=\langle 1\rangle,$

and

$\langle 1\rangle=Z_{0}(G)<Z_{1}(G)<...<Z_{m-2}(G)<Z_{m-1}(G)=G.$

The two series coincide, and the sections are all of order $p$, except the
first which has order $p^{2}$ and is not cyclic. Clearly $G/K_{i}(G)$ is also
of maximal class for $4\leq i\leq m-1$.

######  Proposition 3.1.

Let $G$ be a p-group of maximal class and order $p^{m}$. Then:

  1. (1)

$|G:G^{\prime}|=p^{2}$, $|Z(G)|=p$ and $|K_{i}(G):K_{i+1}(G)|=p$ for $2\leq
i\leq m-1$.

  2. (2)

If $1\leq i<m-1$, then $G$ has only one normal subgroup of order $p^{i}$. More
precisely, if $N$ is a normal subgroup of $G$ of index $p^{i}\geq p^{2}$, then
$N=K_{i}(G)$.

  3. (3)

$G$ has $p+1$ maximal subgroups.

###### Proof.

(1) We have that

$p^{m}=|G|=|G:G^{\prime}|\prod_{i=2}^{m-1}|K_{i}(G):K_{i+1}(G)|$

. Now it suffices to observe that $|G:G^{\prime}|\geq p^{2}$, by theorem 2.4,
and that $|K_{i}(G):K_{i+1}(G)|\geq p$ for $2\leq i\leq m-1$.

(2) Let $N$ be any normal subgroup of $G$ of index $p^{i}$ with $0\leq i\leq
m$.If $i=0$ or $1$ then $N=K_{1}(G)$ or $N$ is maximal in $G$. Otherwise
$i\geq 2$ and $K_{i}(G)\leq N$ by Corollary 2.5. Since $|G:K_{i}(G)|=p^{i}$,
we conclude that $N=K_{i}(G)$.

(3) As $G/K_{2}(G)$ has exponent $p$, the Frattini subgroup
$\Phi(G)=K_{2}(G)$. Hence $G/\Phi(G)$ has order $p^{2}$ and can be regarded as
a vector space over $\mathbb{F}_{p}$ of dimension $2$. This vector space has
$p+1$ subspaces of dimension $1$ and these correspond to the maximal subgroups
of $G$. ∎

######  Remark 3.2.

  1. (1)

(The converse of the proposition 3.1(2)) If $G$ is a noncyclic group of order
$p^{m}>p^{2}$ containing only one normal subgroup of order $p^{i}$ for each
$1\leq i<m-1$, then it is of maximal class [1, Lemma 9.1].

  2. (2)

A p-group $G$ of order $p^{n}$ has exactly $n+p-1$ nontrivial normal subgroups
if and only if it is of maximal class [1, exercie 0.30].

Suppose that a p-group $G$ has only one normal subgroup $T$ of index $\geq
p^{p+1}$. If $G/T$ is of maximal class so is $G$ (see [1, Theorem 12.9]).
Conversely, let $G$ be a p-group of maximal class. If $N$ is a normal subgroup
of $G$ of index $\geq p^{2}$, then $G/N$ has also maximal class. Indeed, since
the class of $G/K_{i}(G)$ is $i-1$ Whenever $2\leq i\leq m$, then the result
follows immediately from proposition 3.1.

######  Proposition 3.3.

Let $G$ be a nonabelian group of order $p^{m}>p^{p}$. If $G$ contains only one
normal subgroup of index $p^{k}$ for any $k\in\\{2,...,p+1\\}$, then it is of
maximal class.

###### Proof.

Obviously, $|G:G^{\prime}|=p^{2}$ hence $d(G)=2$. Assume that $G$ is not of
maximal class, then $m>p+1$. Let $T<G^{\prime}$ be G-invariant of index
$p^{p+1}$ in $G$, then $G/T$ is of maximal class. As, by hypothesis, $T$ is
the unique normal subgroup of index $p^{p+1}$ in $G$, then $G$ is of maximal
class. ∎

######  Proposition 3.4.

(M.Suzuki). Let $G$ be nonabelian p-group. If $A<G$ of order $p^{2}$ is such
that $C_{G}(A)=A$, then $G$ is of maximal class.

###### Proof.

We use induction on $|G|$. Since $p^{2}\nmid|Aut(A)|$ then, by N/C-theorem,
$N_{G}(A)$ is nonabelian of order $p^{3}$. As $A<G$, then $Z(G)<A$ and
$|Z(G)|=p$. Obviously, $C_{G/Z(G)}(A/Z(G))=N_{G/Z(G)}(A/Z(G))$. Since
$C_{G/Z(G)}(N_{G}(A)/Z(G))\leq C_{G/Z(G)}(A/Z(G))=N_{G}(A)/Z(G)$ is of order
$p^{2}$ then, by induction, $G/Z(G)$ is of maximal class so $G$ is also of
maximal class since $|Z(G)|=p$. ∎

######  Corollary 3.5.

A p-group $G$ is of maximal class if and only if $G$ has an element with
centralizer of order $p^{2}$.

######  Proposition 3.6.

Let $G$ be a p-group, $B\leq G$ nonabelian of order $p^{3}$ and $C_{G}(B)<B$.
Then $G$ is of maximal class.

###### Proof.

Assume that $|G|\geq p^{4}$ and the proposition has been proved for groups of
order $<|G|$ . It is known that a Sylow p-subgroup of $Aut(B)$ is nonabelian
of order $p^{3}$. Now, $C_{G}(B)=Z(B)=Z(G)$. Therefore, by N/C-Theorem,
$N_{G}(B)/Z(G)$ is nonabelian of order $p^{3}$. If $x\in G-C_{G}(B)$
centralizes $N_{G}(B)/Z(G)$, then $x$ normalizes $B$ so $x\in N_{G}(B)$, a
contradiction. Thus, $C_{G}(N_{G}(B)/Z(G))<N_{G}(B)/Z(G)$ so, by induction,
$G/Z(G)$ is of maximal class. Since $|Z(G)|=p$, we are done.∎

######  Proposition 3.7.

Let $G$ be a p-group. If $G$ has a subgroup $H$ such that $N_{G}(H)$ is of
maximal class, then it is of maximal class.

###### Proof.

Assume that $|N_{G}(H)|>p^{3}$ (otherwise, $C_{G}(H)=H$ and $G$ is of maximal
class, by Proposition 3.4). We use induction on $|G|$. One may assume that
$N_{G}(H)<G$, then $H$ is not characteristic in $N_{G}(H)$ so by Proposition
3.1, we have $|N_{G}(H):H|=p$ hence $|H|>p$. As $|Z(N_{G}(H))|=p$ and
$Z(G)<N_{G}(H)$, we get $Z(G)=Z(N_{G}(H))$ so $|Z(G)|=p$ and $Z(G)<H$. Then
$N_{G/Z(G)}(H/Z(G))=N_{G}(H)/Z(G)$ is of maximal class, so $G/Z(G)$ is also of
maximal class by induction. Since $|Z(G)|=p$, then $G$ is of maximal class. ∎

######  Lemma 3.8.

Let $G$ be a p-group and let $N\lhd G$ be of order $>p$. Suppose that $G/N$ of
order $>p$ has cyclic center. If $R/N\lhd G/N$ is of order $p$ in $G/N$, then
$R$ is not of maximal class.

###### Proof.

Let $T$ be a G-invariant subgroup of index $p^{2}$ in $N$. Then $R\leq
C_{G}(N/T)$ so $R/T$ is abelian of order $p^{3}$, and we conclude that $R$ is
not of maximal class. ∎

######  Proposition 3.9.

Let $A<G$ be of order $>p$. If all subgroups of $G$ containing $A$ as a
subgroup of index $p$ are of maximal class, then $G$ is also of maximal class.

###### Proof.

Set $N=N_{G}(A)$. In view of proposition 3.7 and hypothesis, one may assume
that $|N:A|>p$ (otherwise, there is nothing to prove). Let $D<A$ be
N-invariant of index $p^{2}$ ( $D$ exists since $|A|>p$). Set $C=C_{N}(A/D)$,
then $C>A$. Let $F/A\leq C/A$ be of order $p$, then $F$ is not of maximal
class, a contradiction. ∎

Let $H<G$ be of index $>p^{k}$, $k>1$. If all subgroups of $G$ of order
$p^{k}|H|$, containing $H$, are of maximal class, then $G$ is also of maximal
class. Indeed, let $H<M<G$, where $|M:H|=p^{k}-1$. Then all subgroups of $G$
containing $M$ as a subgroup of index $p$, are of maximal class. Now the
result follows from Proposition 3.9.

######  Proposition 3.10.

Let $G$ be a p-group of maximal class and order $p^{m}$, $p>2$, $m>3$, and let
$N\lhd G$ be of index $p^{3}$. Then $exp(G/N)=p$.

###### Proof.

Assume that this is false. Let $T$ be a G-invariant subgroup of index $p$ in
$N$. By hypothesis, $G/N$ has two distinct cyclic subgroups $C/N$ and $Z/N$ of
order $p^{2}$. Then $C/N\cap Z/N=Z(G/T)$ and $G/T$ is not of maximal class, a
contradiction. ∎

Let $A$ be an abelian subgroup of index $p$ of a nonabelian p-group $G$. By
[1, lemma 1.1], we have $|G|=p|G^{\prime}||Z(G)|$. Hence, we have the
following proposition.

######  Proposition 3.11.

Suppose that a nonabelian p-group $G$ has an abelian subgroup $A$ of index
$p$. If $|G:G^{\prime}|=p^{2}$, then $G$ is of maximal class.

###### Proof.

We proceed by induction on $|G|$. One may assume that $|G|>p^{3}$. We have
$|Z(G)|=\frac{1}{p}|G:G^{\prime}|=p$, so $Z(G)$ is a unique minimal normal
subgroup of $G$. Since $Z(G)<G^{\prime}$, then wa have
$|G/Z(G):G^{\prime}/Z(G)|=|G:G^{\prime}|=p^{2}$. Hence, the quotient group
$G/Z(G)$ is of maximal class by induction, and the result follows since
$|Z(G)|=p$. ∎

The previous result also holds if $G$ contains a subgroup of maximal class and
index $p$ (see [1, Theorem 9.10]).

######  Exercice 3.1.

Let $G$ be a p-group of order $p^{4}$. Show that $G$ is of maximal class if
and only if $|G:G^{\prime}|=p^{2}$.

Solution. Let $|G:G^{\prime}|=p^{2}$. We have to prove that $G$ is of class
$3$. Assume that $cl(G)=2$. It follows from [2, Lemma 65.1] that $G$ contains
a nonabelian subgroup $B$ of order $p^{3}$. By proposition 3.6, we have
$G=BZ(G)$, then $|G:G^{\prime}|=p^{3}$, a contradiction.

######  Exercice 3.2.

Let $G$ be a p-group of order $p^{4}$ and exponent $p$. Prove that if $G$ has
no nontrivial direct factors then it is of maximal class.

Solution. Let $H$ be an $A_{1}$-subgroup of $G$; then $|H|=p^{3}$. If
$Z(G)\nleq H$, then $G=H\times C$, where $C<Z(G)$ is of order $p$ such that
$C\nleq H$. Thus, $Z(G)<H$ so $Z(G)$ is of order $p$. Since $C_{G}(H)=Z(G)$,
we get $C_{G}(H)<H$. Then, by proposition 3.6, $G$ is of maximal class.

######  Exercice 3.3.

If $G$ is of order $p^{4}$ and exponent $p$. Prove that if $d(G)=2$ then $G$
is of maximal class.

Solution. Obviously, the group $G$ has an abelian subgroup of index $p$. Since
$G^{\prime}=\Phi(G)$ and, by hypothesis, $|G:G^{\prime}|=p^{2}$, the result
follows from Proposition 3.11.

##  4\. Fundamental subgroup of a p-group of maximal class

Let $G$ be a $p$-group of maximal class and order $p^{n}$, $n>3$. The
structure of $G$ is largely determined by the structure of a certain subgroup
$G_{1}$ called the fundamental subgroup of $G$ and defined as follows:
$G_{1}\diagup K_{4}(G)$ is the centralizer of $K_{2}(G)\diagup K_{4}(G)$ in
$G\diagup K_{4}(G)$, where $G>K_{2}(G)$ $>...>K_{n}(G)=1$ is the lower central
series of $G$. For each $i=2,...,m-2$, $K_{i}(G)/K_{i+2}(G)$ is a noncentral
normal subgroup of order $p^{2}$ in ${G}/{K_{i+2}(G)}$ so
$M_{i}=C_{G}(K_{i}(G)/K_{i+2}(G))$ is a maximal subgroup of $G$. Indeed, by
N/C-theorem, the quotient group $G\diagup M_{i}$ is isomorphic to a subgroup
of $\ Aut(K_{i}(G)\diagup K_{i+2}(G))$. But, a p-sylow subgroup of $\
Aut(K_{i}(G)\diagup K_{i+2}(G))$ has order $p$. Thus,
$\left|G,M_{i}\right|=p$.

######  Proposition 4.1.

Let $G$ be a p-group of maximal class. Then $G_{1}$ is a characteristic
subgroup of $G$.

###### Proof.

Let $\varphi\in Aut(G)$. The subgroup $G_{1}$ is composed of the elements
$x\in G$ such that $[x,G_{2}]\leq G_{4}$. The subgroups $G_{2}$ and $G_{4}$
are characteristic in $G$, then we have that

$[\varphi(x),G_{2}]=[\varphi(x),\varphi(G_{2})]=\varphi([x,G_{2}])\leq\varphi(G_{4})=G_{4}.$

This proves that $G_{1}$ is characteristic in $G$. ∎

We shall write $G_{i}$ instead $K_{i}(G)$ when there is no possible confusion.
We have the following main properties:

  1. (1)

$G\diagup G_{2}$ is elementary abelian of order $p^{2}$ and
$\left[G_{i},G_{i+1}\right]=p$ for $1\leq i\leq n-1.$ Hence
$\left|G:G_{i}\right|=p^{i}$ for $1\leq i\leq n.$

  2. (2)

For $i\geq 2$, $G_{i}$ is the unique normal subgroup of index $p^{i}$.

  3. (3)

$G_{i}=Z_{n-i}(G)$, $i=2,...,n$.

######  Remark 4.2.

  1. (1)

If $N$ is a normal subgroup of $G$ such that $|G/N|\geq p^{4}$, it is clear
from the definition that $(G/N)_{1}=G_{1}/N$.

  2. (2)

Let $G$ be a p-group of maximal class, $|G|>p^{4}$. Let $M$ be a maximal
subgroup of $G$ and let $M_{1}$ be the fundamental subgroup of $M$. Then
$|G:M_{1}|=p^{2}$ and $M_{1}\lhd G$ so $M_{1}=\Phi(G)<G_{1}$, and we get
$M_{1}=G_{1}\cap M$.

######  Lemma 4.3.

[1, Theorem 9.6(e)] Let $G$ be a group of maximal class and order $p^{m}$,
$p>2$, $m>p+1$. Then the set of all maximal subgroups of $G$ is

$\Gamma_{1}=\\{M_{1}=G_{1};M_{2};...;M_{p+1}\\}$

Where $G_{1}$ is the fundamental subgroup of $G$, and the subgroups
$M_{2};...;M_{p+1}$ are of maximal class.

######  Proposition 4.4.

Let a p-group $G$ of maximal class have order $p^{m}>p^{p+1}$. If $H<G$ is of
order $>p^{p}$ and $H\nleq G_{1}$, then $H$ is of maximal class.

###### Proof.

We proceed by induction on $m$. If $m=p+2$, the result follows from Lemma 4.3
(indeed, then all members of the set $\Gamma_{1}-{G_{1}}$ are of maximal
class). Now let $m>p+2$ and let $H\leq M\in\Gamma_{1}$, then $M$ is of maximal
class (Lemma 4.3). The subgroup $M_{1}=M\cap G_{1}$ is the fundamental
subgroup of $M$ (Remark 4.2(2)). As $H\nleq M_{1}$, then $H$ is of maximal
class, by induction, applied to the pair $H<M$. ∎

######  Proposition 4.5.

Let $G$ be of maximal class and order $>p^{p+1}$. If $H<G$ is of order
$>p^{2}$, then either $H\leq G_{1}$ or $H$ is of maximal. class.

###### Proof.

Let $|H|=p^{k}$. The result is known if $k=3$ [5]. Assuming that $k>3$, we use
induction on $k$. Then all maximal subgroups of $H$ which $\neq H\cap G_{1}$
are of maximal class, by induction. Then the set $\Gamma_{1}(H)$ contains
exactly $|\Gamma_{1}(H)|-1\neq 0(modp^{2})$ members of maximal class so $H$ is
of maximal class, by [1, Theorem 12.12(c)]. ∎

######  Exercice 4.1.

[1, Exercise 9.28] Let $G$ be a p-group of maximal class and order $>p^{p+1}$.
Show the following:

  1. (1)

$exp(G_{1})=exp(G)$.

  2. (2)

If $x\in G$ is of order $\geq p^{3}$, then $x\in G_{1}$.

## References

  * [1] Y. Berkovich, Groups of Prime Power Order, Vol. 1, Walter de Gruyter · Berlin · New York, 2008.
  * [2] Y.Berkovich and Z. Janko, Groups of Prime Power Order, Vol. 2, Walter de Gruyter, Berlin, 2008.
  * [3] Y. Berkovich and Z. Janko, Groups of Prime Power Order, Vol. 3 , Walter de Gruyter, Berlin, 2011.
  * [4] Y. Berkovich, Some remarks on subgroups of maximal class of a finite $p$-group, J. Algebra and Applications 14, 1550043 (2015).
  * [5] N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45-92.
  * [6] M. E. Charkani and N. Snanou, On a special class of finite p-groups of maximal class and exponent p, JP Journal of Algebra, Number Theory and Applications 44(2) (2019):251-260.
  * [7] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967.
  * [8] C. R. Leedham-Green and S. McKay, On p-groups of maximal class, I, Quart. J. Math. Oxford (2) 27 (1976), 297-311.
  * [9] C. R. Leedham-Green and S. McKay, The structure of groups of prime power order, London Mathematical Society Monographs. New Series, vol. 27, Oxford University Press, Oxford, 2002.
  * [10] J.J. Rotman, An Introduction to the Theory of Groups, 4th ed., Springer-Verlag, New York, 1995.
  * [11] A. Wiman, Über mit Diedergruppen verwandte p-Gruppen, Arkiv för Matematik, Astronomi och Fysik 33A (1946), 1-12.
  * [12] A. Wiman, Über p-Gruppen von maximaler Klasse, Acta Math. 88 (1952), 317-346.

Noureddine Snanou
1 Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed
Ben Abdellah University, Fez, Morocco
Email<EMAIL_ADDRESS>