# A Role of Fractional Dimension in Study Physics

###### Abstract

We study physics from a fractional dimensional perspective. It is considered
that the whole space is not only integer dimensions but also a superposition
of spaces between integer dimensions. Space-filling between integer dimensions
makes a fractional dimensional space. This happens through a rotation of the
integer dimension by the fractional derivative operator (local part) to sweep
all spaces between two integer dimensions. Then, we introduce four axioms
which they can help us to build a mathematical framework for study physics
from fractional dimension point of view. Afterward, three scenarios such as
static, linear and quadratic trajectories are demonstrated. Where moving with
linear or quadratic trajectories in fractional dimensional space corresponds
to an inertial framework or accelerated system in physics, respectively. Also
from this opinion, it is realized that the coupling of space and time (space-
time) happens through the dimension with an interconnection role where we call
it space-dimension-time. Finally, the wave equation is derived from fractional
dimensional point of view with a linear trajectory in fractional dimensional
space. And the behavior of electromagnetic wave in lossless and lossy mediums,
Doppler effect and gravitational red (blue)-shift are also demonstrated from
fractional dimensional aspect.

††journal: osac††articletype: Research Article

Ali Dorostkar
Fractal Group, Isfahan, Iran
<EMAIL_ADDRESS>

##  1 Introduction

Perception of the nature is one of the main topics of science and particularly
in study of physics. Based on the Gödel’s incompleteness theorems [1], a
feasibility of complete understanding of the nature is impossible. It means
that there is no and will not have a complete and comprehensive theory for the
nature and we can just improve a model in each step and this process will
continue for ever. This process has continued during the history of physics.
The classical or Newtonian’s physics as a fundamental framework of physics is
a great tool to describe most of phenomena in the nature. However, the lack of
solution for justification of particles dynamic in microscopic scale physics
or a reason for gravitational force has excited us to go further step to bring
quantum mechanics (QM) and relativity. As we know from general relativity (GR)
being created, gravity is due to the curvature of space-time by mass which
presents a large scale physics [2]. On the other side, quantum mechanics is a
good model for small scale physics [3]. However, there are some open questions
such as trajectory of particle and non-locality effect in QM. There are also
several attempts to unify a theory for physics through a relation between GR
and QM such as M-theory or string theory[4, 5], however; there is still lack
of connection between quantum and gravity. This introduction reminds us there
are always some open questions in physics leading the best understanding of
nature. In this regard, to understand how more information about physics can
be derived from fractional dimensional point of view, firstly we express our
axioms and then we get started to make a concept: how does a moving of
particle in the integer space correspond to that in the fractional dimensional
space? For instance , three axes of $x_{1},x_{2},x_{3}$ make a space of three
integer dimensions. While in our perspective, in addition of integer
dimensions the fractional dimensions must be taken into account. Fractional
basis function (vector) can be expressed by a rotation of basis function
(vector) of integer dimensions [6]. It is shown that this rotation can be
described by the fractional derivative (FDr) operator. Then, this rotation
fills the whole space between the integer dimensions as the fractional
dimensional space. We express several axioms and subsequently make a
mathematical framework through local FDr operator. Afterward, we can
investigate moving (propagating) a particle (signal) through particular
trajectories in fractional dimensional space.

##  2 Axioms

The general framework of the proposed model in fractional dimension for
physics study has been built based on the following axioms.
1)  Every phenomenon in nature can be presented by a wave function $(\Psi(S)$)
in fractional dimensional space. Where fractional dimensional space is the
whole space between and included linear integer spaces dimensions.Where S as
indicated in (1) is a vector of independent variables of space (
$x_{i}(i=1,2,3)or{\bf r}$) and time (t).

$\begin{split}S=[x_{1},x_{2},x_{3},t].\end{split}$ (1)

2) Every phenomenon in nature is a combination of local and nonlocal parts,
with the local parts being a good approximation for classical physics.

3) The final wave function ($\Psi$) can be calculated by taking the
normalized fractional derivative of the initial wave function ($\Psi_{0}$)
through a trajectory in fractional dimensional space.

$\begin{split}\Psi=D_{\rm N}^{\alpha(S)}\Psi_{0},\end{split}$ (2)

where $D_{\rm N}$ and $\alpha(S)$ are normalized fractional derivative
operator and fractional order corresponding to the trajectory of the moving
system in fractional dimensional space, respectively. Also, it can be
described by considering postulate 2 as follows:

$D_{\rm N}^{\alpha(S)}\Psi_{0}=\rm{local\ term+nonlocal\ term}.$ (3)

In most physics problems, we can consider the following good approximation:

$D_{\rm N,L}^{\alpha(S)}\Psi_{0}\approx\rm{local\ term},$ (4)

where index $\\{\rm N,L\\}$ means considering the local part of FDr with
normalization.

4) Dimension Principle: Dimensions of an object will be smaller (larger) with
increasing (decreasing) the spatial distance. It is assumed as a principle for
fractional dimensional point of view where it is related to a projective space
originated from perspective visual effect[7, 8]. Thus, the equivalent
statement can be expressed as follows:
let the observer in S coordinate and the object in $S^{\prime}$ coordinate,
then there is an acute angle between them which is the angle is zero when the
coordinate distance is zero and will be 90 degrees when the distance goes to
infinity (or where the object disappears from the observer). As will show,
this rotation can be described through FDr operator. As shown in Fig. 1, an
object in $S^{\prime}$ coordinate will be smaller and smaller when it is
moving away from an observer in $S$ coordinate. This can be realized through a
trajectory of $S^{\prime}$ coordinates where $S^{\prime}$ is not orthogonal
toward $S$ coordinate and makes a particular angle in each space-time
position. In other words, the projection of coordinate $S^{\prime}$ on $S$
will change depending on the distance. As it is shown mathematically in axiom
3, the dimension variation is following through a fractional derivative with a
certain trajectory. Note that, in Fig. 1 the scalar 1 is derived from an
integer derivative of x or t.

Figure 1: Schematic of S and $S^{\prime}$ coordinates in a movement framework

##  3  Mathematical Formulation

In this section, we make a mathematical formulation for the fractional
dimensional basis vector and function. Before further processing, we
demonstrate some important FDr definitions and determine a local FDr
expression for some special functions in this study. Noted that in this study
we have started with an approximation of the local part
($D^{\alpha}\Psi_{0}\approx\rm{local\ term}$) to build a mathematical
framework for physics with fractional dimension point of view.

###  3.1  Definitions and a general framework

We describe an applicable mathematical tool for the purposes of this study by
definitions of the basis functions for fractional domain by the realization of
linear and fractional spaces.
Definition 1
Fractional space is a whole space between and included linear integer spaces.

Definition 2
If suppose the fractional space is a space between two consecutive integer
dimensions, then the basis functions in the fractional space can be
represented as a rotation of basis functions of the linear space.

Definition 3
The fractional derivative operator ($D_{t}^{\alpha}$) is a nonlocal operatore
and there are several definitions for FDr such as direct, Riemann-Liouville
(RL), Caputo (C), Liouville–Caputo and Fourier methods[9, 10]. The definition
for RL and C FDr is expressed as follows

$\begin{split}&{}_{a}^{RL}D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}(\frac{d}{dt})^{n}\int_{a}^{t}(t-x)^{n-\alpha-1}f(x)dx,\\\
&{}_{a}^{C}D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{a}^{t}(t-x)^{n-\alpha-1}\frac{d^{n}f(x)}{dx^{n}}dx,\\\
\end{split}$ (5)

where $D^{\alpha}$, a, $\Gamma$ are the FDr operator of order $\alpha$, an
initial point for integral and Gamma function, respectively.

Definition 4
The fractional derivative has two terms one common term and one extra term
which is different depends on the type of FDr definition.

${}_{a}D_{t}^{\alpha}f(t)=\rm{common\ term+extra\ term=local\ term+nonlocal\
term}$.

There is an extra term as well as common one for every definitions presented
for FDr based on the generalized hypergeometric function [10]. The common term
is called a local fractional derivative because it can also be derived from an
induction of integer derivative. Therefore, we call it local FDr.

Definition 5
The normalized FDr is a division of FDr over a normalization factor. Where
normalization factor is an amplitude of FDr which is a function of $\alpha$.
This makes confidence the normality of the basis functions holds.

Hint: For sake of simplicity and as a first investigation of fractional
dimension in study physics, we continue with normalized local FDr part. The
nonlocal part of fractional derivative can help us to explain a non-locality
effect in physics. It is under further investigation and we will add more
results about that in the future.

Normalized local FDr of some important functions
In this study, we need FDr of trigonometric, exponential and power series
functions for further investigations. In this regard, we use the local term of
FDr definition which is common for all definitions or from direct method by
induction. Therefore, the following results are derived by a normalization of
local FDr (LFDr) through a division by a normalization factor. More details
are in the appendix.

$\begin{split}&D_{\rm N,L}^{\alpha}\bigg{\\{}{cos(\omega
S)\bigg{\\}}}=\frac{1}{D_{0}}D^{\alpha}\bigg{\\{}{cos(\omega
S)\bigg{\\}}}=\frac{1}{D_{0}}(\omega)^{\alpha}cos(\omega
S+\frac{\pi}{2}\alpha)=cos(\omega S+\frac{\pi}{2}\alpha),\\\ &D_{\rm
N,L}^{\alpha}\bigg{\\{}{sin(\omega
S)\bigg{\\}}}=\frac{1}{D_{0}}D^{\alpha}\bigg{\\{}{sin(\omega
S)\bigg{\\}}}=\frac{1}{D_{0}}(\omega)^{\alpha}sin(\omega
S+\frac{\pi}{2}\alpha)=\sin(\omega S+\frac{\pi}{2}\alpha),\\\ &D_{\rm
N,L}^{\alpha}\bigg{\\{}{e^{(i\omega
S)}\bigg{\\}}}=\frac{1}{D_{0}}D^{\alpha}\bigg{\\{}{e^{(i\omega
S)}\bigg{\\}}}=\frac{1}{D_{0}}(\omega)^{\alpha}e^{(\omega
S+\frac{\pi}{2}\alpha)}=e^{i(\omega S+\frac{\pi}{2}\alpha)},\end{split}$ (6)

where $D_{\rm N,L}^{\alpha}$ and $D_{\rm 0}$ denote the normalized local FDr
operator and the normalization factor, respectively. This make confidence the
normality of the basis functions holds. $D_{0}$ denotes $(\omega)^{\alpha}$.
Local fractional derivative of a basic power function, $x^{n}$, can be derived
from direct or Riemann–Liouville or Caputo or other methods with the same
result[13]

$D_{\rm
N,L}^{\alpha}\bigg{\\{}{x^{n}\bigg{\\}}}=\frac{1}{D_{0}}D^{\alpha}\bigg{\\{}{x^{n}\bigg{\\}}}=\frac{1}{D_{0}}\frac{\Gamma(n+1)x^{(n-\alpha)}}{\Gamma(n-\alpha+1)}=x^{(n-\alpha)},$
(7)

where $D_{0}$ is equal to $\frac{\Gamma(n-\alpha+1)}{\Gamma(n+1)}$.

Theorem 1
Let normalized LFDr operator applying in power series $(x^{n})$ and
trigenometric functions ($e^{(i\omega S)}$), then this operator is a
commutative and symmetry for these functions as follows:

$D_{\rm N,L}^{\alpha_{1}}D_{\rm N,L}^{\alpha_{2}}=D_{\rm
N,L}^{\alpha_{1}+\alpha_{2}},$ (8) $D_{\rm N,L}^{\alpha}I_{\rm
N,L}^{\alpha}=\mathds{1},$ (9)

where the proof can be easily made for these functions as shown in the
appendix.

####  3.1.1 Fractional Dimensional Basis Vector

Let $S$ and $S_{\alpha}$ are the basis vector of the linear and fractional
spaces, respectively. Then, the relation between two coordinates regarding (7)
is written as follows

$S_{\alpha}=D_{\rm N,L}^{\alpha}\bigg{\\{}{S\bigg{\\}}}={S^{(1-\alpha)}}.$
(10)

In conclusion, the coordinate $S_{\alpha}$ (red arrow) for a certain
${\alpha}$ as shown in Fig. 2 is a rotation of original coordinate S in
fractional dimensional space. It can be derived from a LFDr operator with a
certain order of ${\alpha}$.

Figure 2: Schematic of fractional dimensional basic vector $S_{\alpha}$ (red
arrow) from a rotation of integer basis vector S through FDr operator, a
vector rotates from 0 dimension to 1 ($0\leq\alpha\leq 1$) or zero to 90
degrees rotation ($0\leq\theta\leq\frac{\pi}{2}$).

In Fig. 3, it is shown how a basic vector x through a rotation can reach 1.
Scalar 1 is chosen because an integer derivative of x is equal to one. This
will happen via a fractional derivative with the angle
$\theta=\frac{\pi}{2}\alpha$. We need three orthogonal axes such as $x$
(representation of $x_{i}$), $t$ and $1$ as shown in Fig. 3(a) for depicting
the fractional space of space-time, where two of them including $x$, $t$ are
vector and $1$ denotes scalar one. Red colour arrows depicted in Fig. 3(b) to
(d) show that space-time axes for $\alpha$ near to zero (Fig. 3(b)) mostly has
a angle ($\theta$) near to zero, since $\alpha$ goes to one (Fig. 3(d)) the
angle ($\theta$) is near to $\pi/2$.

Figure 3: Rotation process of vector axes x and t through fractional space.
(a) all vector axes are orthogonal ($\theta=0$ or $\alpha=0$), then by
movement, the vector axes x and t will rotate: (b) $\theta\approx 0$
($\alpha\approx 0$) , (c) $0<\theta<\pi/2$ ($0<\alpha<1$) and (d)
$\theta\approx\pi/2$ ($\alpha\approx 1$).

As will shown in the next section, there are different scenarios depending on
the angle rotation of $\theta$.

####  3.1.2 Fractional Dimensional Basis Function

Let $\phi(S)$ and $\phi_{\alpha}(S)$ be the basis functions of the linear and
fractional spaces, respectively. Then, there are the following relations
between the basis functions of both spaces:

$\phi_{\alpha}(S)=D_{\rm N}^{\alpha}\bigg{\\{}{\phi(S)\bigg{\\}}}.$ (11)

Note that $\alpha$ is a certain order of fractional dimension for each space-
time position.

##  4 Preliminary plan

We limit our study to the three scenarios as shown in Table 1 due to having
more general cases in physics. Note that most of the problems in physics can
be covered by a trajectory with a maximum of quadratic order in fractional
dimensional space for a moving system, as shown in Table 1.
The first is called static scenario when $\alpha$ is a constant value. The
latter is a linear dynamic that corresponds to an inertial framework in
physics. It will happen when $\alpha$ has a linear trajectory. The last is an
accelerated system when the quadratic trajectory function acts. In this
section, first of all, we discuss our attitude regarding dimension and then we
investigate these three scenarios for both fractional dimensional basis
vectors and functions. The trajectory function coefficients of $\alpha_{i}$,
$\beta_{ij}$ and $\gamma_{ij}$ shown in Table 1 can be expressed in the format
of a vector or matrix (tensor) depend on the trajectory of the fractional
dimension.

Table.1:Different Observations
---
Scenario | Static | Linear Dynamic | Accelerated Dynamic
Trajectory ($\alpha_{i}(S)$) | $\alpha_{i}$ | $\beta_{ij}S_{i}+\alpha_{i}$ | $\gamma_{ijk}S_{j}S_{k}+\beta_{ij}S_{i}+\alpha_{i}$

###  4.1 Space-Dimension-Time

As we know, the concept of space-time comes from relativity theory and
expresses that they coupled to each other for a moving system. However, there
is a question of why they have coupled with each other even though it seems
that they are independent. In this approach, there is an interconnection
between space and time through the dimension we call it Space-Dimension-Time
(SDT). In other words, space-time have been coupled to each other through the
dimension where in Fig. 4 shows this interconnection. Although space and time
are two independent variables, however; dimension through a dynamical movement
interconnect them to each other.

Figure 4: Space-time coupling through the dimension called Space-Dimension-
Time (SDT)

There is no coupling in static scenario state since it is a steady one. To
show a coupling between space and time, one could consider both of them as
simple linear movement. Indeed, $x_{i}$ and t axes move rotationally toward
scalar 1 axis. Being independent space and time asserts to us to write as
follows

$\begin{split}&\alpha_{i}(x_{i})=\beta_{i}x_{i}+\alpha_{i}\;\;(i=1,2,3),\\\
&\alpha_{4}(t)=\beta_{4}t+\alpha_{4},\end{split}$ (12)

taking an integer derivative

$\begin{split}&d\alpha_{i}(x_{i})=\beta_{i}dx_{i},\\\
&d\alpha_{4}(t)=\beta_{4}dt,\end{split}$ (13)

and then by dividing of them, we have

$\begin{split}\frac{d(\alpha_{i}(x_{i}))}{d(\alpha_{4}(t))}=\frac{\beta_{i}dx_{i}}{\beta_{4}dt}=\frac{\beta_{i}}{\beta_{4}}v_{i}.\end{split}$
(14)

The relation (14) says us when moving with a constant speed, movement of
rotational variable x-axis toward t one has a relation to speed. In other
words, space is related to time when looking at it from dimensional point of
view.

###  4.2  Presentation of Static and Moving Coordinate Systems via Fractional
Dimensional Basis Vector

Besides static case, there are two important types of dynamic systems in
physics either movement of constant velocity or acceleration. These types of
systems in our study correspond to an angle rotation of coordinate
$S_{\alpha}$ of a linear rotation (constant velocity) or quadratic rotation
(constant acceleration). It should be noted that if two coordinates of $S$ and
$S_{\alpha}$ are lying with each other, then $\alpha$ is equal to zero or
$\theta=0$ and when they are so far from each other $\alpha$ is going to 1
($\theta\to\frac{\pi}{2}$). This property is independent of the type of
dynamic system. It means that S and $S_{\alpha}$ are lying each other for both
accelerated and inertial framework systems when $\alpha$ is equal to zero and
when two coordinates are faring away for both systems $\alpha$ is going to 1.

####  4.2.1 Static scenario

In this case, two coordinates S and $S_{\alpha}$ are considered to be fixed.
Therefore, there is a constant space distance ($S_{\rm offset}$) between a
fixed observer and static framework. It should be noted that the space
distance offset is equal to $|S_{\rm offset}|=\sqrt{x_{1,\rm
offset}^{2}+x_{2,\rm offset}^{2}+x_{3,\rm offset}^{2}}$ and time distance
offset can be zero for static case, however; it is not valid for dynamic
scenario because we have space-time distance or in this aspect SDT distance.
To find $\alpha$, it is needed to define an infinite space parameter
($|S_{\infty}|$). Infinite space (time) parameter ($|S_{\infty}|$) is a
certain value of space (time) where after that the observer (camera) can not
measure the observed framework anymore and it disappears from the observer
point of view
($|S_{\rm\infty}|=\sqrt{x_{1,\rm\infty}^{2}+x_{2,\rm\infty}^{2}+x_{3,\rm\infty}^{2}}$).
Therefore, $\alpha$ for a static scenario can be realized as follows

$\alpha=\frac{|S_{\rm offset}|}{|S_{\infty}|}=\alpha_{0}.$ (15)

So, the fractional dimensional basis vector is

$S_{\alpha}=D_{\rm N,L}^{\alpha_{0}}\big{\\{}{S\big{\\}}}.$ (16)

Different situations for static observation are tabulated in Table 1. When two
coordinates are lying each other the $S_{\rm offset}$ is equal to zero
resulting $\alpha=0$ ($S=S_{\alpha}$). If $S_{\rm offset}$ has a value larger
than zero and smaller than $S_{\infty}$, thus we have an $\alpha_{0}$ in a
range between zero and one ( $0<\alpha_{0}<1$) and when two coordinates are
very far each other $S_{\rm offset}\to S_{\infty}$ which leads to $\alpha\to
1$. For instance, imagine one object is in front of the observer, then $S_{\rm
offset}=0$ and therefore $\alpha=\alpha_{0}=0$. Now, we put the same object
with a distance of $S_{\rm offset}=\frac{1}{2}S_{\infty}$ and then from the
same observer the $\alpha=\alpha_{0}=0.5$. It must be mentioned when $S_{\rm
offset}>S_{\infty}$ the measurement is not correct due to disappearing the
observed framework from the observer and leads to high error tolerance.

Table.1:Different Situations for Static Observation
---
$S_{\rm offset}$ | $S_{\rm offset}=0$ | $0<S_{\rm offset}<S_{\infty}$ | $S_{\rm offset}\to S_{\infty}$ | $S_{\rm offset}>S_{\infty}$
$\alpha=\alpha_{0}$ | $\alpha_{0}=0$ | $0<\alpha_{0}<1$ | $\alpha_{0}\to 1$ | $\alpha_{0}>1$
Measurement | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\times$

####  4.2.2 Linear and Quadratic Dynamic

A Moving system leads to a rotation of coordinate depends on the function
trajectory in space-time from fractional dimensional point of view. To sake of
simplicity, linear and quadratic trajectories of moving coordinate are chosen.
Then, we find an equivalent physical problem for that. In physics, when we
have inertial framework (constant speed), special relativity and Minkowski
space are used, and when we have accelerated frames (constant acceleration or
gravity), general relativity and Riemann space are used. With a physical
counterpart, we should obtain the same result with special relativity when
$\alpha$ has a linear trajectory and general relativity when $\alpha$ has a
quadratic trajectory.

Further investigations need a new metric for fractional dimensional space-time
or SDT. It is out of this study and we will consider it as a future work,
however; we try to express an equivalency of our model with some simplified
examples. We can represent a $S^{\prime}$ coordinate based on the
trigonometric properties as following

$\begin{split}\begin{pmatrix}x_{1}^{\prime}\\\ x_{2}^{\prime}\\\
x_{3}^{\prime}\\\
t^{\prime}\end{pmatrix}=\begin{pmatrix}cos\big{(}\theta_{1}(S)\big{)}&0&0&0\\\
0&cos\big{(}\theta_{2}(S)\big{)}&0&0\\\
0&0&cos\big{(}\theta_{3}(S)\big{)}&0\\\
0&0&0&cos\big{(}\theta_{4}(S)\big{)}\\\ \end{pmatrix}\begin{pmatrix}x_{1}\\\
x_{2}\\\ x_{3}\\\
t\end{pmatrix}+\begin{pmatrix}sin\big{(}\theta_{1}(S)\big{)}\\\
sin\big{(}\theta_{2}(S)\big{)}\\\ sin\big{(}\theta_{3}(S)\big{)}\\\
sin\big{(}\theta_{4}(S)\big{)}\end{pmatrix},\end{split}$ (17)

where $\theta_{i}(S)$ (i=1,2,3,4) is equal to $\frac{\pi}{2}\alpha_{i}(S)$. As
an example we consider a situation for very slow motion of an inertial
framework in direction of $x_{1}$ where $\theta_{1}$ is very small
($cos\theta_{1}\approx 1$ and $sin\theta_{1}\approx\theta_{1}$) and
$\theta_{2}=\theta_{3}=\theta_{4}=0$. Therefore the coordinate transformation
can be approximated to Galilean transformation with $sin\theta_{1}\approx\pm
vt$.

$\begin{split}\begin{pmatrix}x_{1}^{\prime}\\\ x_{2}^{\prime}\\\
x_{3}^{\prime}\\\ t^{\prime}\end{pmatrix}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\\\ x_{3}\\\
t\end{pmatrix}+\begin{pmatrix}\pm vt\\\ 0\\\ 0\\\ 0\end{pmatrix}.\end{split}$
(18)

Similarly, for an accelerated system in $x_{1}$ or $x^{\prime}_{1}$ direction
with a small acceleration or gravity and an approximation of
$sin\theta_{1}\approx\theta_{1}=\pm\frac{1}{2}gt^{2}$ and
$\theta_{2}=\theta_{3}=\theta_{4}=0$, the coordinate transformation can be
expressed as follows

$\begin{split}\begin{pmatrix}x_{1}^{\prime}\\\ x_{2}^{\prime}\\\
x_{3}^{\prime}\\\ t^{\prime}\end{pmatrix}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\\\ x_{3}\\\
t\end{pmatrix}+\begin{pmatrix}\pm\frac{1}{2}gt^{2}\\\ 0\\\ 0\\\
0\end{pmatrix}.\end{split}$ (19)

###  4.3  A Signal Representation of System via Fractional Dimensional Basis
Function

In this section, we demonstrate the signal behavior based on the three
scenarios. By the use of assumption (11), we can also rewrite the formulation
between basis function of transmitter ($T(S)$) and receiver ($R(S)$) signals
as follows

$R(S)=\frac{1}{2}D_{\rm
N}^{\alpha}\big{\\{}{T(S)\big{\\}}}+\frac{1}{2}\bigg{(}D_{\rm
N}^{\alpha}\big{\\{}{T(S)\big{\\}}}\bigg{)}^{*}=Re\bigg{\\{}{D_{\rm
N}^{\alpha}\big{\\{}{T(S)\big{\\}}}\bigg{\\}}},$ (20)

where $\Re$ is real part of basis function. The FDr operator connects the
input signal (initial signal) to the output signal of the system through a
trajectory in fractional dimension. This assumption denotes that the
transmitter itself is in an integer coordinate, however; from receiver point
of view has a fractional dimensional coordinate.
Note that for the sake of simplicity we play with two variables $x_{1}$
(space) and t (time) where $x_{1}$ is symbolized by $z$. So hereafter, $z$ and
t denote space and time variables, respectively.

####  4.3.1  Static scenario

From signal point of view , a constant distance holds between transmitter and
receiver as shown in Fig. 5 (a). This is like that an observer see a standing
object from a certain fixed distance.

Figure 5: Schematic of three scenarios (a) static ($v=0$) (b) Linear dynamic
($v=v_{0}$) (c) accelerated system ($a=a_{0}$)

For example, a signal of formula $A_{0}cos(\omega t+kz)$ is propagating in
free space . How is the received signal diagram change $(R(z=L,t))$ after a
propagation of distance L in a (a) lossless (b) lossy medium? Where $A_{0}$,
$\omega$ and $k$ are amplitude, angular frequency and wave vector,
respectively.
Hint: when electromagnetic wave is propagating in a lossy medium, wave vector
is a complex value as

$k=\beta+i\alpha_{att.},$ (21)

where $\beta$ and $\alpha_{att.}$ are propagation constant and attenuation
decay coefficient.
The answer for part (a) is:
from electrodynamics side, we know that the signal has a phase shift
corresponding to $kL$ where in lossless medium $k$ is equal to propagation
constant $\beta$.

$R(z=L,t)=A_{0}cos(\omega t+\beta L)=A_{0}cos(\omega t+\Phi),$ (22)

where $\Phi$ is equal to $\beta L$.
However from fractional dimensional point of view, the received signal at
$z=L$ is obtained by a constant fractional dimension of $\alpha_{0}$.
Therefore, we have

$R(z=L,t)=D_{\rm N,L}^{\alpha_{0}}\big{\\{}{A_{0}cos(\omega
t)\big{\\}}}=A_{0}cos(\omega t+\frac{\pi\alpha_{0}}{2})=A_{0}cos(\omega
t+\Phi),$ (23)

where $\Phi=\beta L=\frac{\pi\alpha_{0}}{2}$.
and the answer for part (b) is:
from electrodynamics aspect, the final received signal after a simplification
at the distance L is

$R(z=L,t)=A_{0}e^{-\alpha_{att.}L}cos(\omega t+\beta L),$ (24)

and from fractional dimensional perspective, we have

$R(z=L,t)=\frac{1}{2}D_{\rm N,L}^{\alpha_{0}}\big{\\{}{e^{i\omega
t}\big{\\}}}+\frac{1}{2}\bigg{(}D_{\rm N,L}^{\alpha_{0}}\big{\\{}{e^{i\omega
t}\big{\\}}}\bigg{)}^{*}=\frac{1}{2}D_{\rm
N,L}^{\alpha_{0}}\big{\\{}{e^{i\omega t}\big{\\}}}+\frac{1}{2}\bigg{(}D_{\rm
N,L}^{\alpha^{*}_{0}}\big{\\{}{e^{-i\omega t}\big{\\}}}\bigg{)},$ (25)

where $\alpha_{0}$ has a complex value as follows

$\alpha_{0}=\alpha^{\rm R}_{0}+i\alpha^{\rm I}_{0}.$ (26)

Then, the complex fractional basis function is

$D_{\rm N,L}^{\alpha_{0}}\big{\\{}{e^{i\omega t}\big{\\}}}=e^{i(\omega
t+\frac{\pi\alpha_{0}}{2})}=e^{i(\omega t+\frac{\pi}{2}\alpha^{\rm
R}_{0}+i\frac{\pi}{2}\alpha^{\rm I}_{0})}=e^{(-\frac{\pi}{2}\alpha^{\rm
I}_{0})}e^{i(\omega t+\frac{\pi}{2}\alpha^{\rm R}_{0})},$ (27)

where $D_{\rm N,L}^{\alpha_{0}^{*}}\big{\\{}{e^{-i\omega t}\big{\\}}}$ is
equal to the complex conjugate of $D_{\rm
N,L}^{\alpha_{0}}\big{\\{}{e^{i\omega t}\big{\\}}}$. Thus, after a
rearrangement we have

$R(z=L,t)=A_{0}\Re\bigg{\\{}{D_{\rm N,L}^{\alpha_{0}}(\big{\\{}{e^{i\omega
t}\big{\\}}}\bigg{\\}}}=A_{0}e^{-(\frac{\pi}{2}\alpha^{\rm I}_{0})}cos(\omega
t+\frac{\pi}{2}\alpha^{\rm R}_{0}),$ (28)

It can be observed that decaying a signal as loss in linear vector space
corresponds to a trajectory of signal in imaginary domain of fractional
dimension. Therefore, we have:

$\frac{\pi}{2}\alpha^{\rm R}_{0}=\beta L,$ (29)

and

$\frac{\pi}{2}\alpha^{\rm I}_{0}=\alpha_{att.}L.$ (30)

####  4.3.2  Dynamic Scenario

In this case, at least one of the transmitter or receiver is moving in
fractional dimensions. Therefore, it is called dynamic observation. Here, two
types of movement of linear and quadratic trajectories are demonstrated.

##### 4.3.2.1 Linear dynamic

We show that the two important cases in physics such as wave equation and
Doppler effect are equivalent to a movement system in fractional dimensional
space with a linear trajectory.

######  4.3.2.1.1 Equivalency for the Wave Equation

We start to reformulate a new representation of the wave equation based on the
proposed assumption. As we know, wave is a phenomenon in time which exactly
propagates in the space. It can be corresponded to a temporal (spatial) signal
moving in spatial (temporal) domain with a fractional dimensional trajectory
of $\alpha(\bf r)$ ($\alpha(t)$). The wave function (${\bf\Psi}(\bf{r}$,$t))$
can be decomposed into the two independently variable functions of $t$ and
space variables $\bf{r}$ (${\bf{\Psi}}$
$(\bf{r}$$,t)$$=\psi_{1}(t)\psi_{2}(\bf{r})$). Therefore, the wave equation
can be realized as follows

$D_{\rm N}^{\alpha_{1}(\bf r)}\bigg{\\{}{{\bf\psi_{1}}(t)\bigg{\\}}}=D_{\rm
N}^{\alpha_{2}(t)}\bigg{\\{}{\\{{\bf\psi_{2}}(\bf r)\bigg{\\}}}.$ (31)

The left hand side of (31) means that temporal signal ($\psi_{1}(t)$) moves in
spatial domain with a trajectory of $\alpha_{1}(\bf r)$ and has to be equal to
the right one of equation where it shows the movement of spatial signal
($\psi_{2}(\bf{r})$ in time domain with a trajectory of $\alpha_{2}(t)$. For
sake of simplicity as we know from quantum or electrodynamics, the wave basis
function of two variables $z$ and $t$ (${\bf\Psi}(\bf{z}$,$t)$) is equal to
$e^{i(\omega t+kz)}$. Thus, the fractional dimensional wave equation is

$D_{\rm N,L}^{\alpha_{1}(z)}\bigg{\\{}{e^{i\omega t}\bigg{\\}}}=D_{\rm
N,L}^{\alpha_{2}(t)}\bigg{\\{}{e^{ikz}\bigg{\\}}}=e^{i(\omega
t+\frac{\pi}{2}\alpha_{1}(z))}=e^{i(kz+\frac{\pi}{2}\alpha_{2}(t))}=e^{i(\omega
t+kz)},$ (32)

where trajectories of $\alpha_{i}$ (i=1,2) are

$\begin{split}\alpha_{1}(z)=\frac{2}{\pi}kz\\\
\alpha_{2}(t)=\frac{2}{\pi}\omega t,\end{split}$ (33)

######  4.3.2.1.2 Doppler effect

As shown in Fig.5 (b) for an inertial system, an observer has stayed in a
certain fixed position (here $z=L$) and the transmitter is closing (faring
away) to the receiver with a velocity of $v_{t}$. This phenomenon is well-
known in physics and called the Doppler effect. Here, it is shown how the
movement of the signal source with a linear trajectory in fractional dimension
is equivalent to the Doppler effect. Let us to have a signal, $T(z=0,t)$, at
transmitter of the following expression

$\displaystyle T(z=0,t)=A_{0}cos(\omega t).$ (34)

Therefore, by considering the trajectory as follows

$\alpha(z,t)=\beta_{41}z+\beta_{44}t,$ (35)

where $\beta_{41}$ and $\beta_{44}$ constant coefficients and letting $z$ as a
constant value vs. t, the received signal is

$R(z=L,t)=D_{\rm N,L}^{\alpha(S)}\bigg{\\{}{A_{0}cos(\omega
t)\bigg{\\}}}=A_{0}cos(\omega t+\frac{\pi\alpha(S)}{2})=A_{0}cos(\omega
t+\frac{\pi(\beta_{41}z+\beta_{44}t)}{2}).$ (36)

After a rearrangement, it is

$R(z=L,t)=A_{0}cos(\omega_{D}t+\Phi),$ (37)

where constant phase is

$\begin{split}\Phi=\frac{\pi}{2}\beta_{41}L,\end{split}$ (38)

and $\omega_{D}$ is a new (Doppler) angular frequency.

$\begin{split}\omega_{D}=\omega+\frac{\pi\beta_{44}}{2}=2\pi(f+\Delta f)=2\pi
f_{D},\end{split}$ (39)

with Doppler frequency shift [16]

$\begin{split}f_{D}=\bigg{(}\frac{c\pm v_{r}}{c\pm
v_{t}}\bigg{)}f=f+\bigg{(}\frac{\pm v_{r}\mp v_{t}}{c\pm
v_{t}}\bigg{)}f=f+\Delta f,\end{split}$ (40)

where c is the propagation speed of wave in the medium. When the receiver is
moving towards (away) the source, the sign is positive (negative). For this
scenario observer velocity ($v_{r}$) is zero ($v_{r}=0$), and $\Delta f$ is:

$\begin{split}\Delta f=\bigg{(}\frac{\mp v_{t}}{c\pm
v_{t}}\bigg{)}f=\frac{\beta_{44}}{4}.\end{split}$ (41)

####  4.3.3 Acceleration dynamic scenario

In this case, trajectory in fractional dimension is a quadratic function.
Here, we assumed the following trajectory

$\alpha(S)=\alpha(z,t)=\gamma_{41}z^{2}+\gamma_{44}t^{2}+\beta_{41}z+\beta_{44}t,$
(42)

where $\beta_{41}$, $\beta_{44}$, $\gamma_{41}$ and $\gamma_{44}$ constant
coefficients where with zeroing the $\gamma$ coefficients the Doppler problem
will be obtained. Therefore, the basis function for accelerated system can be
presented as follows

$\phi^{\alpha(z,t)}(S)=D_{\rm
N}^{(\gamma_{41}z^{2}+\gamma_{44}t^{2}+\beta_{41}z+\beta_{44}t)}\bigg{\\{}{\phi(S)\bigg{\\}}},$
(43)

and the received signal regarding being constant $z$ vs. t is

$\begin{split}&R(z,t)=D_{\rm N,L}^{\alpha(z,t)}\bigg{\\{}{A_{0}cos(\omega
t)\bigg{\\}}}=A_{0}cos(\omega t+\frac{\pi\alpha(z,t)}{2})\\\ &=A_{0}cos(\omega
t+\frac{\pi(\gamma_{41}z^{2}+\gamma_{44}t^{2}+\beta_{41}z+\beta_{44}t)}{2}).\end{split}$
(44)

After a rearrangement, it is

$R(z=L,t)=A_{0}cos(\omega_{Da}(t)t+\Phi),$ (45)

where constant phase at $z=L$ is

$\begin{split}\Phi=\frac{\pi}{2}(\gamma_{14}L^{2}+\beta_{14}L)\end{split}$
(46)

and $\omega_{Da}(t)$ is a new (Doppler) angular time dependent as follows

$\begin{split}\omega_{Da}(t)=\frac{\pi\gamma_{44}}{2}t+\omega+\frac{\pi\beta_{44}}{2}=\frac{\pi\gamma_{44}}{2}t+\omega_{D}.\end{split}$
(47)

It is corresponded with the gravitational red (blue)-shift. As a simple
example, the profile of (47) with all coefficients normalized to one is
depicted in Fig. 6. It can be observed that its profile shows a gravitational
red (blue)-shift behavior [17] and also without consideration of envelope, it
has a similar behavior like gravitational wave [18].

Figure 6: The plot of cosine function with quadratic trajectory in fractional
dimension.

##  5 Acknowledgement

The author is very appreciate for fruitful discussion with A. Sabihi and his
valuable comments.

## References

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  * [12] M. Chen, S. Shao, and P. Shi, Robust adaptive control for fractional-order systems with disturbance and saturation (John Wiley and Sons, 2017).
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  * [16] V. C. Chen, The micro-Doppler effect in radar (Artech house, 2019).
  * [17] A. Eddington, “Einstein shift and doppler shift,” Nature 117, 86–86 (1926).
  * [18] W. G. Anderson and J. D. Creighton, Gravitational-wave physics and astronomy: An introduction to theory, experiment and data analysis (John Wiley and Sons, 2012).

##  6 Appendix

###  6.1 Proof Theorem 1

Let normalized LFDr operator applying in power series $(x^{n})$ and
trigenometric function ($e^{(i\omega S)}$), then this operator is a
commutative and symmetry for these function as follows:

$D_{\rm N,L}^{\alpha_{1}}D_{\rm N,L}^{\alpha_{2}}=D_{\rm
N,L}^{\alpha_{1}+\alpha_{2}},$ (48) $D_{\rm N,L}^{\alpha}I_{\rm
N,L}^{\alpha}=\mathds{1},$ (49)

Proof
We can start with cosine function and applying in the left hand side of Eq.
(48):

$D_{\rm N,L}^{\alpha_{1}}D_{\rm N,L}^{\alpha_{2}}\bigg{\\{}{cos(\omega
S)\bigg{\\}}}=D_{\rm N,L}^{\alpha_{1}}\bigg{\\{}{cos(\omega
S+\frac{\pi}{2}\alpha_{2})\bigg{\\}}}=cos(\omega
S+\frac{\pi}{2}\alpha_{2}+\frac{\pi}{2}\alpha_{1}),$ (50)

then for the right side we have :

$D_{\rm N,L}^{\alpha_{1}+\alpha_{2}}\bigg{\\{}{cos(\omega
S)\bigg{\\}}}=cos(\omega S+\frac{\pi}{2}(\alpha_{1}+\alpha_{2})).$ (51)

Therefore, they are equivalent and with the same analogy it is consistent for
the sinus , complex basis and power series functions.
For the proof of Eq. (49), firstly we have to mentioned that the fractional
integral of order $\alpha$ ($I_{\rm N,L}^{\alpha}$) of an arbitrary function
$f$ can be expressed based on the FDr.

$I_{\rm N,L}^{\alpha}f=D_{\rm N,L}^{-\alpha}f.$ (52)

Then, we have

$D_{\rm N,L}^{\alpha}I_{\rm N,L}^{\alpha}\bigg{\\{}{cos(\omega
S)\bigg{\\}}}=D_{\rm N,L}^{\alpha}\bigg{\\{}{cos(\omega
S-\frac{\pi}{2}\alpha)\bigg{\\}}}=cos(\omega
S+\frac{\pi}{2}\alpha-\frac{\pi}{2}\alpha)=cos(\omega S),$ (53)

and it can be applied for the other proposed functions.

###  6.2  Fractional Dimensional Basis Vector

There is another approach to reach the same result as presented in Eq. (10).
When the relation between two coordinates of S and $S_{\alpha}$ can be
expressed based on the superposition of FDr (order of $\alpha$) of S
coordinate in one hand and the fractional integral (order of $\beta=1-\alpha$)
of 1 on the other hand. It should be noted that summation of order of
fractional operators has to be one since it should correspond to 90 degree
rotation.

$\begin{split}S_{\alpha}=(1-\alpha)D_{\rm N,L}^{\alpha}\\{{S\\}}+\alpha I_{\rm
N,L}^{\beta}\\{{1\\}}=(1-\alpha)D_{\rm N,L}^{\alpha}\\{{S\\}}+\alpha D_{\rm
N,L}^{-\beta}\\{{1\\}}.\end{split}$ (54)

Where the coefficients of $D{\rm N,L}^{\alpha}$ and $I{\rm N,L}^{\alpha}$ as
weight factors are $(1-\alpha)$ and $(\alpha)$, respectively. These weight
factors have been considered for the satisfaction of rotation coordinate when
the trajectory ($\alpha(S)$) is equal to zero both coordinates of $S$ and
$S_{\alpha}$ are lying each other and when $\alpha$ is equal to one, it leads
to 90 degree rotation of S coordinate in fractional dimension space and will
be 1. Therefore, $S_{\alpha}$ coordinate is

$\begin{split}S_{\alpha}=(1-\alpha)S^{1-\alpha}+\alpha
S^{1-\alpha}=S^{1-\alpha}.\end{split}$ (55)

###  6.3  Example

For more clarification, the 3-D plot of FDr of function f(x)=x for an interval
x between [0,10] and $\alpha$ between [0,1] are depicted in Fig.6.

Figure 7: FDr of function f(x)=x for interval of [0,10] and alpha between zero
and one.

We wish to see choosing a trajectory on the red line in Figure 6 how the FDr
diagram does change? The red line corresponds to a trajectory of $\alpha=.1x$
which leads to the FDr plot on the proposed trajectory as shown in Fig. 7. It
means for every point of x, there is a special $\alpha$ which makes a function
of $\alpha(x)$.

Figure 8: FDr of function f(x)=x for interval of [0,10] for the trajectory of
$\alpha=.1x$. It should be noted that for every point of x we have a special
constant value of $\alpha$.


# MCMC to address model misspecification in Deep Learning classification of
Radio Galaxies

Devina Mohan
Department of Physics & Astronomy
University of Manchester, UK
<EMAIL_ADDRESS>
Anna Scaife
Department of Physics & Astronomy
University of Manchester, UK
<EMAIL_ADDRESS>
The Alan Turing Institute, 96 Euston Rd, London, UK<EMAIL_ADDRESS>
###### Abstract

The radio astronomy community is adopting deep learning techniques to deal
with the huge data volumes expected from the next-generation of radio
observatories. Bayesian neural networks (BNNs) provide a principled way to
model uncertainty in the predictions made by deep learning models and will
play an important role in extracting well-calibrated uncertainty estimates
from the outputs of these models. However, most commonly used approximate
Bayesian inference techniques such as variational inference and MCMC-based
algorithms experience a "cold posterior effect (CPE)", according to which the
posterior must be down-weighted in order to get good predictive performance.
The CPE has been linked to several factors such as data augmentation or
dataset curation leading to a misspecified likelihood and prior
misspecification. In this work we use MCMC sampling to show that a Gaussian
parametric family is a poor variational approximation to the true posterior
and gives rise to the CPE previously observed in morphological classification
of radio galaxies using variational inference based BNNs.

##  1 Introduction

The next-generation of radio astronomy facilities such as the Square Kilometre
Array (SKA) will produce huge volumes of data and the use of deep learning
(DL) methods is inevitable given the expected data volumes [1, 2]. Modern
astrophysics is driven by population analyses and any automated classification
pipeline should produce well-calibrated uncertainty estimates that quantify
the model uncertainty introduced in the results. In this work we consider the
morphological classification of radio galaxies and discuss the challenges
faced while implementing Bayesian Convolutional Neural Networks (CNNs) for
their classification.

While several works have looked at classifying radio galaxies with deep
learning [e.g. 3, 4, 5, 6, 7], with the exception of [8] and [9], little work
has been done on understanding the degree of confidence with which CNN models
predict the class of individual radio galaxies. In general, it has been
suggested that deep learning models produce overconfident predictions [10] and
provide no uncertainty estimates, which are essential for scientific
application of these models. On the other hand, probabilistic models such as
Bayesian neural networks (BNNs) provide a principled way to model uncertainty
[11, 12] by specifying priors, $P(\theta)$, over the neural network
parameters, $\theta$, and learning the posterior distribution, $P(\theta|D)$,
over those parameters, where $D$ is the data.

Recovering this posterior distribution directly is intractable for neural
networks. Several techniques have been developed to approximate Bayesian
inference for neural networks among which Variational Inference (VI) and Monte
Carlo (MC) Dropout methods are most commonly used. VI assumes an approximate
posterior from a family of tractable distributions, and converts the inference
problem into an optimisation problem [13, 14, 15]. The model learns the
parameters of the distributions by minimising an Evidence Lower Bound
Objective (ELBO) function.

Another easily implemented Bayesian approximation is MC Dropout, which learns
a distribution over the network outputs by setting randomly selected weights
of the network to zero with probability, $p$ [16]. MC dropout can be
considered an approximation to VI, where the variational approximation is a
Bernoulli distribution. Although a convenient technique, this method lacks
flexibility and does not fully capture the uncertainty in model predictions,
especially under covariate shift where the data distributions at training and
test time are not identically distributed [17].

However, there are several challenges in implementing BNNs in practice.
Several published works have reported that their BNNs experience a "cold
posterior effect (CPE)", according to which the posterior needs to be down-
weighted or tempered with a temperature term, $T\leq 1$, in order to get good
predictive performance [18]:

$P(\theta|D)\propto(P(D|\theta)P(\theta))^{1/T}.$ (1)

Previous work has shown that VI based BNN models experience a CPE when
classifying radio galaxies [9]. The choice of variational approximation limits
the variational posterior to specific regions of the true posterior density
space and it is difficult to evaluate how good the variational approximation
is without having access to the true posterior [19]. Several hypothesis have
been put forward to explain the CPE including likelihood, prior and model
misspecification [18, 20, 21].

In this work we demonstrate that, for radio galaxy classification, using MCMC
to recover posterior distributions on neural network parameters suggests that
the "cold posterior effect" previously observed with VI models is due to model
misspecification arising from poor variational posterior approximations. We
also compare model performance for different approximate Bayesian inference
methods including VI and MC Dropout and present preliminary uncertainty
calibration results.

##  2 MCMC for Neural Networks

MCMC methods are a class of algorithms used to obtain samples from probability
distributions which are otherwise intractable or do not have a full analytical
description. The first application of MCMC to neural networks was proposed by
[22], who introduced Hamiltonian Monte Carlo (HMC) from quantum chromodynamics
to the general statistics literature. However, it wasn’t until [23] introduced
Stochastic Gradient Langevin Dynamics (SGLD), that MCMC for neural networks
became feasible for large datasets. More recently, [24] have revisited HMC and
proposed novel data splitting techniques to make it work with large datasets.
We use the HMC algorithm in our work.

Hamiltonian Monte Carlo HMC simulates the path of a particle traversing the
negative posterior density space using Hamiltonian dynamics [25, 26, 27]. To
apply HMC to deep learning, the neural network parameter space is augmented by
specifying an additional momentum variable, $m$, for each parameter, $\theta$.
Therefore, for a $d$-dimensional parameter space, the augmented parameter
space contains $2d$ dimensions. We can then define a log joint density as
follows:

$\mathrm{log}[p(\theta,m)]=\mathrm{log}[p(\theta|D)p(m)]\,.$ (2)

Hamiltonian dynamics allows us to travel on the contours defined by the joint
density of the position and momentum variables, also known as the phase space.
The Hamiltonian function is given by:

$H(\theta,m)=U(\theta)+K(m)=constant,$ (3)

where $U(\theta)$ is the potential energy and $K(m)$ is the kinetic energy.
The potential energy is defined to be the negative log posterior probability
and the kinetic energy is usually assumed to be quadratic in nature and of the
form $K(m)=(1/2)\,m^{T}M^{-1}m$, where $M$ is a positive-definite mass matrix.
This corresponds to the negative probability density of a zero-mean Gaussian,
$p(m)=\mathcal{N}(m|0,M)$, with covariance matrix, M, which is usually assumed
to be the identity matrix.

The partial derivatives of the Hamiltonian describe how the system evolves
with time. In order to solve the partial differential equations using
computers, we need to discretise the time, $t$, of the dynamical simulation
using a step-size, $\epsilon$. The state of the system can then be computed
iteratively at times $\epsilon$, $2\epsilon$, $3\epsilon$… and so on, starting
at time zero upto a specified number of steps, $L$. The leapfrog integrator is
used to solve the system of partial differential equations. Two
hyperparameters, the step-size, $\epsilon$, and the number of leapfrog steps,
$L$, together determine the trajectory length of the simulation. The partial
derivative of the potential energy with respect to the position, $\partial
U/\partial\theta$, can be calculated using the automatic differentiation
capabilities of most standard neural network libraries.

In each iteration of the HMC algorithm, new momentum values are sampled from
Gaussian distributions, followed by simulating the trajectory of the particles
according to Hamiltonian dynamics for $L$ steps using the leapfrog integrator
with step-size $\epsilon$. At the end of the trajectory, the final position
and momentum variables, $(\theta^{*},m^{*})$, are accepted based on a
Metropolis-Hastings accept/reject criterion that evaluates the Hamiltonian for
the proposed parameters and the previous parameters.

##  3 Experimental Setup

Data The MiraBest dataset used in this work consists of 1256 images of radio
galaxies of $150\times 150$ pixels pre-processed to be used specifically for
deep learning tasks [28]. The galaxies are labelled using the FRI and FRII
morphological types based on the definition of [29] and further divided into
their subtypes. In addition to labelling the sources as FRI, FRII and their
subtypes, each source is also flagged as ‘Confident’ or ‘Uncertain’ to
indicate the human classifiers’ confidence while labelling the dataset. In
this work we use the MiraBest Confident subset and consider only the binary
FRI/FRII classification. The training and validation sets are created by
splitting the predefined training data into a ratio of 80:20. The final split
consists of 584 training samples, 145 validation samples, and 104 withheld
test samples. No data augmentation is used.

Architecture We use an expanded LeNet-5 architecture with two additional
convolutional layers with 26 and 32 channels, respectively, to be consistent
with the literature on using BNNs for classifying the MiraBest dataset [9].
The model has $232,444$ parameters in total.

MCMC Inference We use the hamiltorch
package111https://github.com/AdamCobb/hamiltorch developed by [24] for scaling
HMC to large datasets. Using their HMC sampler, we set up two HMC chains of
$200,000$ steps using different random seeds and run it on the MiraBest
Confident dataset. We use a step size of $\epsilon=10^{-4}$ and set the number
of leapfrog steps to $L=50$. We specify a Gaussian prior over the network
parameters and evaluate different prior widths,
$\sigma=\\{{1,10^{-1},10^{-2},10^{-3}\\}}$, using the validation data set. We
find that $\sigma={10^{-1}}$ results in the best predictive performance and
consequently use it to define the prior width for all weights and biases of
the neural network in our experiments. A burn-in of $20,000$ samples is
discarded. To compute the final posteriors we thin the chains by a factor of
1000 to reduce the autocorrelation in the samples and obtain 180 samples. A
compute time of $150$ hrs is required to run the inference on two Nvidia A100
GPUs. The Gelman-Rubin diagnostic, $\hat{R}$, is used to assess the
convergence of our HMC chains [30]. If $\hat{R}\approx 1$ we consider the MCMC
chains for that particular parameter to have converged. While $\hat{R}$ values
for some parameters in the network are greater than $1$, the final two neurons
in the last layer of our network have $\hat{R}\leq 1$. We also monitor the
negative log-likelihood and accuracy, which converge by the
$100,000^{\textit{\emph{th}}}$ inference step.

Other models For the VI implementation we use a Gaussian variational
approximation to the posterior and consider different priors including
Gaussian and Laplace distributions following [9]. The Laplace prior provides
optimal predictive performance and lowest uncertainty calibration error,
however for direct comparison to our HMC baseline we also consider a Gaussian
prior with $\sigma=0.01$. Results are reported for a tempered VI posterior,
with $T=0.01$ in Table 1. For the MC Dropout model, a dropout rate of 50% is
implemented before the last layer of our neural network, which is standard for
CNNs [8, 16]. The network is trained for 150 epochs using the Adam optimser
with a learning rate of $10^{-3}$ and a weight decay of $10^{-4}$. We obtain
200 samples from VI and MC Dropout posterior predictive distributions by
passing each sample in the test set through the test loop 200 times. We use
the same optimiser hyperparameters as the MC Dropout training for our non-
Bayesian CNN model. A compute time of $12$ mins is required to train the VI
model on a single Nvidia A100 GPU.

Code for this work is available at https://github.com/devinamhn/RadioGalaxies-
BNNs

##  4 Results

Cold posterior effect Previous work on using VI for radio galaxy
classification has shown that the "cold posterior effect" (CPE) persists even
when the learning strategy is modified to compensate for model
misspecification with a second order PAC-Bayes bound to improve the
generalisation performance of the network [9, 31], see Figure 1. We do not
observe a CPE when we use samples from our HMC inference to construct the
posterior predictive distribution for classifying the MiraBest dataset (orange
dashed line in Figure 1). This suggests that using a Gaussian parametric
family as a variational approximation to the true posterior distribution is a
poor assumption and leads to a misspecified model, which gives rise to the
CPE. In the general Bayesian DL literature, some authors argue that CPE is
mainly an artifact of data augmentation [20], while others have shown that
data augmentation is a sufficient but not necessary condition for CPE to be
present [21]. We find that data augmentation does not have a significant
effect on our HMC and VI models.

Figure 1: The “cold posterior” effect (CPE) observed in VI models (solid blue
line) persists inspite of a PAC-Bayes correction term to account for model
misspecification (red dashed line) when classifying radio galaxies. Using
samples from HMC, we demonstrate that using a Gaussian variational
distribution leads to a poor approximation to the posterior, giving rise to
the CPE. Data are also shown for MC Dropout (green dot-dashed line) and our
non-Bayesian CNN (solid gray line) for comparison.

Model performance The test error is calculated by taking an average of the
predictions obtained using the expected value of the posterior predictive
distribution for each galaxy in the MiraBest Confident test set for different
models, see Table 1. No data augmentation is used during training/inference.
The non-Bayesian CNN and MC Dropout perform comparably in terms of test error.
HMC is more accurate than VI, but does not match the predictive performance of
MC Dropout. We note that it is not performance alone that is important for our
application, but also the calibration of the posterior uncertainties which
will influence the scientific analysis performed using the catalogues
generated by DL pipelines. We have conducted preliminary uncertainty
calibration experiments using the $64\%$ credible intervals of the posterior
predictive distributions to calculate the class-wise expected Uncertainty
Calibration Error (cUCE) values for the predictive entropy [9, 16, 32].
However, at this stage we do not draw any strong conclusions from the
uncertainty quantification experiments, which will be considered more fully in
our future work.

Table 1: Test error and class-wise expected uncertainty calibration error (cUCE) for predictive entropy are reported for different BNNs for the MiraBest Confident dataset. Model | Error (%) | % cUCE
---|---|---
Non-Bayesian CNN | $3.33$ | $\times$
MC Dropout | $3.85\pm 0.19$ | $9.75$
VI (Laplace prior) | $10.58\pm 0.30$ | $14.26$
VI (Gaussian prior) | $12.96\pm 0.33$ | $30.05$
HMC | $6.73\pm 0.25$ | $13.17$

##  5 Conclusions

Using samples from HMC, we find that the cold posterior effect previously
observed in the morphological classification of radio galaxies using
variational inference arises from using a misspecified parametric family to
approximate the posterior. While MCMC does not provide the most
computationally efficient framework for approximate Bayesian inference for
neural networks, it produces asymptotically exact samples from the posterior
which are useful for developing more accurate approximate Bayesian inference
techniques for the radio galaxy classification problem in future.

## Acknowledgments and Disclosure of Funding

AMS gratefully acknowledges support from an Alan Turing Institute AI
Fellowship EP/V030302/1.

## References

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# NILPOTENT RESIDUAL
OF A FINITE GROUP

Eliana Rodrigues  Department of Academic Areas, Instituto Federal de Goiás,
Formosa-GO 73813-816, Brazil<EMAIL_ADDRESS>,  Emerson de Melo
Department of Mathematics, University of Brasília, Brasília-DF 70910-900,
Brazil<EMAIL_ADDRESS>and  Gülin Ercan  Department of Mathematics, Middle
East Technical University, 06800, Ankara/Turkey<EMAIL_ADDRESS>
###### Abstract.

Let $F$ be a nilpotent group acted on by a group $H$ via automorphisms and let
the group $G$ admit the semidirect product $FH$ as a group of automorphisms so
that $C_{G}(F)=1$. We prove that the order of $\gamma_{\infty}(G)$, the rank
of $\gamma_{\infty}(G)$ are bounded in terms of the orders of
$\gamma_{\infty}(C_{G}(H))$ and $H$, the rank of $\gamma_{\infty}(C_{G}(H))$
and the order of $H$, respectively in cases where either $FH$ is a Frobenius
group; $FH$ is a Frobenius-like group satisfying some certain conditions; or
$FH=\langle\alpha,\beta\rangle$ is a dihedral group generated by the
involutions $\alpha$ and $\beta$ with $F=\langle\alpha\beta\rangle$ and
$H=\langle\alpha\rangle$.

###### Key words and phrases:

Frobenius groups, Frobenius-like groups, Dihedral groups, Automorphisms,
Nilpotent residual

###### 1991 Mathematics Subject Classification:

20D45

##  1\. Introduction

Throughout all groups are finite. Let a group $A$ act by automorphisms on a
group $G$. For any $a\in A$, we denote by $C_{G}(a)$ the set $\\{x\in
G:x^{a}=x\\},$ and write $C_{G}(A)=\bigcap_{a\in A}C_{G}(a).$ In this paper we
focus on a certain question related to the strong influence of the structure
of such fixed point subgroups on the structure of $G$, and present some new
results when the group $A$ is a Frobenius group or a Frobenius-like group or a
dihedral group of automorphisms.

In what follows we denote by $A^{\\#}$ the set of all nontrivial elements of
$A$, and we say that $A$ acts coprimely on $G$ if $(|A|,|G|)=1$. Recall that a
Frobenius group $A=FH$ with kernel $F$ and complement $H$ can be characterized
as a semidirect product of a normal subgroup $F$ by $H$ such that $C_{F}(h)=1$
for every $h\in H^{\\#}$. Prompted by Mazurov’s problem $17.72$ in the
Kourokva Notebook [26], some attention was given to the situation where a
Frobenius group $A=FH$ acts by automorphisms on the group $G$. In the case
where the kernel $F$ acts fixed-point-freely on $G$, some results on the
structure of $G$ were obtained by Khukhro, Makarenko and Shumyatsky in a
series of papers [8], [9], [10], [11], [12], [13], [14]. They observed that
various properties of $G$ are in a certain sense close to the corresponding
properties of the fixed-point subgroup $C_{G}(H)$, possibly also depending on
$H$. In particular, when $FH$ is metacyclic they proved that if $C_{G}(H)$ is
nilpotent of class $c$, then the nilpotency class of $G$ is bounded in terms
of $c$ and $|H|$. In addition, they constructed examples showing that the
result on the nilpotency class of $G$ is no longer true in the case of non-
metacyclic Frobenius groups. However, recently in [6] it was proved that if
$FH$ is supersolvable and $C_{G}(H)$ is nilpotent of class $c$, then the
nilpotency class of $G$ is bounded in terms of $c$ and $|FH|$.

Later on, as a generalization of Frobenius group the concept of a Frobenius-
like group was introduced by Ercan and Güloğlu in [16], and their action
studied in a series of papers [18], [19],[20],[23],[24],[21]. A finite group
$FH$ is said to be Frobenius-like if it has a nontrivial nilpotent normal
subgroup $F$ with a nontrivial complement $H$ such that $FH/F^{\prime}$ is a
Frobenius group with Frobenius kernel $F/F^{\prime}$ and complement $H$ where
$F^{\prime}=[F,F]$. Several results about the properties of a finite group $G$
admitting a Frobenius-like group of automorphisms $FH$ aiming at restrictions
on $G$ in terms of $C_{G}(H)$ and focusing mainly on bounds for the Fitting
height and related parameters as a generalization of earlier results obtained
for Frobenius groups of automorphisms; and also new theorems for Frobenius-
like groups based on new representation-theoretic results. In these papers two
special types of Frobenius-like groups have been handled. Namely, Frobenius-
like groups $FH$ for which $F^{\prime}$ is of prime order and is contained in
$C_{F}(H)$; and the Frobenius-like groups $FH$ for which $C_{F}(H)$ and $H$
are of prime orders, which we call Type I and Type II, respectively throughout
the remainder of this paper.

In [25] Shumyatsky showed that the techniques developed in [14] can be used in
the study of actions by groups that are not necessarily Frobenius. He
considered a dihedral group $D=\langle\alpha,\beta\rangle$ generated by two
involutions $\alpha$ and $\beta$ acting on a finite group $G$ in such a manner
that $C_{G}(\alpha\beta)=1$. In particular, he proved that if $C_{G}(\alpha)$
and $C_{G}(\beta)$ are both nilpotent of class $c$, then $G$ is nilpotent and
the nilpotency class of $G$ is bounded solely in terms of $c$. In [5], a
similar result was obtained for other groups. It should also be noted that in
[24] an extension of [25] about the nilpotent length obtained by proving that
the nilpotent length of a group $G$ admitting a dihedral group of
automorphisms in the same manner is equal to the maximum of the nilpotent
lengths of the subgroups $C_{G}(\alpha)$ and $C_{G}(\beta)$.

Throughout we shall use the expression “$(a,b,\dots)$-bounded” to abbreviate
“bounded from above in terms of $a,b,\dots$ only”. Recall that the rank
$\mathbf{r}(G)$ of a finite group $G$ is the minimal number $r$ such that
every subgroup of $G$ can be generated by at most $r$ elements. Let
$\gamma_{\infty}(G)$ denote the nilpotent residual of the group $G$, that is
the intersection of all normal subgroups of $G$ whose quotients are nilpotent.
Recently, in [4], de Melo, Lima and Shumyatsky considered the case where $A$
is a finite group of prime exponent $q$ and of order at least $q^{3}$ acting
on a finite $q^{\prime}$-group $G$. Assuming that
$|\gamma_{\infty}(C_{G}(a))|\leq m$ for any $a\in A^{\\#}$, they showed that
$\gamma_{\infty}(G)$ has $(m,q)$-bounded order. In addition, assuming that the
rank of $\gamma_{\infty}(C_{G}(a))$ is at most $r$ for any $a\in A^{\\#}$,
they proved that the rank of $\gamma_{\infty}(G)$ is $(m,q)$-bounded. Later,
in [3], it was proved that the order of $\gamma_{\infty}(G)$ can be bounded by
a number independent of the order of $A$.

The purpose of the present article is to study the residual nilpotent of
finite groups admitting a Frobenius group, or a Frobenius-like group of Type I
and Type II, or a dihedral group as a group of automorphisms. Namely we obtain
the following results.

Theorem A Let $FH$ be a Frobenius, or a Frobenius-like group of Type I or Type
II, with kernel $F$ and complement $H$. Suppose that $FH$ acts on a finite
group $G$ in such a way that $C_{G}(F)=1$. Then

  * a)

$|\gamma_{\infty}(G)|$ is bounded solely in terms of $|H|$ and
$|\gamma_{\infty}(C_{G}(H))|$;

  * b)

the rank of $\gamma_{\infty}(G)$ is bounded in terms of $|H|$ and the rank of
$\gamma_{\infty}(C_{G}(H))$.

Theorem B Let $D=\langle\alpha,\beta\rangle$ be a dihedral group generated by
two involutions $\alpha$ and $\beta$. Suppose that $D$ acts on a finite group
$G$ in such a manner that $C_{G}(\alpha\beta)=1$. Then

  * a)

$|\gamma_{\infty}(G)|$ is bounded solely in terms of
$|\gamma_{\infty}(C_{G}(\alpha))|$ and $|\gamma_{\infty}(C_{G}(\beta))|$;

  * b)

the rank of $\gamma_{\infty}(G)$ is bounded in terms of the rank of
$\gamma_{\infty}(C_{G}(\alpha))$ and $\gamma_{\infty}(C_{G}(\beta))$.

The paper is organized as follows. In Section 2 we list some results to which
we appeal frequently. Section 3 is devoted to the proofs of two key
propositions which play crucial role in proving Theorem A and Theorem B whose
proofs are given in Section 4.

##  2\. Preliminaries

If $A$ is a group of automorphisms of $G$, we use $[G,A]$ to denote the
subgroup generated by elements of the form $g^{-1}g^{a}$, with $g\in G$ and
$a\in A$. Firstly, we recall some well-known facts about coprime action, see
for example [7], which will be used without any further references.

######  Lemma 2.1.

Let $Q$ be a group of automorphisms of a finite group $G$ such that
$(|G|,|Q|)=1$. Then

  * (a)

$G=C_{G}(Q)[G,Q]$.

  * (b)

$Q$ leaves some Sylow $p$-subgroup of $G$ invariant for each prime
$p\in\pi(G)$.

  * (c)

$C_{G/N}(Q)=C_{G}(Q)N/N$ for any $Q$-invariant normal subgroup $N$ of $G$.

We list below some facts about the action of Frobenius and Frobenius-like
groups. Throughout, a non-Frobenius Frobenius-like group is always considered
under the hypothesis below.

Hypothesis* Let $FH$ be a non-Frobenius Frobenius-like group with kernel $F$
and complement $H$. Assume that a Sylow $2$-subgroup of $H$ is cyclic and
normal, and $F$ has no extraspecial sections of order $p^{2m+1}$ such that
$p^{m}+1=|H_{1}|$ for some subgroup $H_{1}\leq H$.

It should be noted that Hypothesis* is automatically satisfied if either
$|FH|$ is odd or $|H|=2$.

######  Theorem 2.2.

Suppose that a finite group $G$ admits a Frobenius group or a Frobenius-like
group of automorphisms $FH$ with kernel F and complement H such that
$C_{G}(F)=1$. Then $C_{G}(H)\neq 1$ and $\mathbf{r}(G)$ is bounded in terms of
$\mathbf{r}(C_{G}(H))$ and $|H|.$

######  Proposition 2.3.

Let $FH$ be a Frobenius, or a Frobenius-like group of Type I or Type II.
Suppose that $FH$ acts on a $q$-group $Q$ for some prime $q$ coprime to the
order of $H$ in case $FH$ is not Frobenius. Let $V$ be a $kQFH$-module where
$k$ is a field with characteristic not dividing $|QH|.$ Suppose further that
$F$ acts fixed-point freely on the semidirect product $VQ$. Then we have
$C_{V}(H)\neq 0$ and

$Ker(C_{Q}(H)\ \textrm{on}\ C_{V}(H))=Ker(C_{Q}(H)\ \textrm{on}\ V).$

###### Proof.

See [17] Proposition 2.2 when $FH$ is Frobenius; [18] Proposition C when $FH$
is Frobenius-like of Type I; and [22] Proposition 2.1 when $FH$ is Frobenius-
like of Type II. It can be easily checked that [17] Proposition 2.2 is valid
when $C_{Q}(F)=1$ without the coprimeness condition $(|Q|,|F|)=1.$ ∎

The proof of the following theorem can be found in [25] and in [2].

######  Theorem 2.4.

Let $D=\langle\alpha,\beta\rangle$ be a dihedral group generated by two
involutions $\alpha$ and $\beta$. Suppose that $D$ acts on a finite group $G$
in such a manner that $C_{G}(\alpha\beta)=1$. Then

  * (a)

$G=C_{G}(\alpha)C_{G}(\beta)$;

  * (b)

the rank of $G$ is bounded in terms of the rank of $C_{G}(\alpha)$ and
$C_{G}(\beta)$;

######  Proposition 2.5.

Let $D=\langle\alpha,\beta\rangle$ be a dihedral group generated by the
involutions $\alpha$ and $\beta.$ Suppose that $D$ acts on a $q$-group $Q$ for
some prime $q$ and let V be a $kQD$-module for a field $k$ of characteristic
different from $q$ such that the group $F=\langle\alpha\beta\rangle$ acts
fixed point freely on the semidirect product $VQ$. If $C_{Q}(\alpha)$ acts
nontrivially on $V$ then we have $C_{V}(\alpha)\neq 0$ and $Ker(C_{Q}(\alpha)\
\textrm{on}\ C_{V}(\alpha))=Ker(C_{Q}(\alpha)\ \textrm{on}\ V)$.

###### Proof.

This is Proposition C in [24]. ∎

The next two results were established in [15, Lemma 1.6] .

######  Lemma 2.6.

Suppose that a group $Q$ acts by automorphisms on a group $G$. If $Q=\langle
q_{1},\ldots,q_{n}\rangle$, then $[G,Q]=[G,q_{1}]\cdots[G,q_{n}].$

######  Lemma 2.7.

Let $p$ be a prime, $P$ a finite $p$-group and $Q$ a $p^{\prime}$-group of
automorphisms of $P$.

  * a)

If $|[P,q]|\leq m$ for every $q\in Q$, then $|Q|$ and $|[P,Q]|$ are
$m$-bounded.

  * b)

If $r([P,q])\leq m$ for every $q\in Q$, then $r(Q)$ and $r([P,Q])$ are
$m$-bounded.

We also need the following fact whose proof can be found in [1].

######  Lemma 2.8.

Let $G$ be a finite group such that $\gamma_{\infty}(G)\leq F(G)$. Let $P$ be
a Sylow $p$-subgroup of $\gamma_{\infty}(G)$ and $H$ be a Hall
$p^{\prime}$-subgroup of $G$. Then $P=[P,H]$.

##  3\. Key Propositions

We prove below a new proposition which studies the actions of Frobenius and
Frobenius-like groups and forms the basis in proving Theorem A.

######  Proposition 3.1.

Assume that $FH$ be a Frobenius group, or a Frobenius-like group of Type I or
Type II with kernel $F$ and complement $H$. Suppose that $FH$ acts on a
$q$-group $Q$ for some prime $q$. Let $V$ be an irreducible
$\mathbb{F}_{p}QFH$-module where $\mathbb{F}_{p}$ is a field with
characteristic $p$ not dividing $|Q|$ such that $F$ acts fixed-point-freely on
the semidirect product $VQ$. Additionaly, we assume that $q$ is coprime to
$|H|$ in case where $FH$ is not Frobenius. Then $\mathbf{r}([V,Q])$ is bounded
in terms of $\mathbf{r}([C_{V}(H),C_{Q}(H)])$ and $|H|$.

###### Proof.

Let $\mathbf{r}([C_{V}(H),C_{Q}(H)])=s.$ We may assume that $V=[V,Q]$ and
hence $C_{V}(Q)=0$. By Clifford’s Theorem, $V=V_{1}\oplus\cdots\oplus V_{t}$,
direct sum of of $Q$-homogeneous components $V_{i}$ , which are transitively
permuted by $FH$. Set $\Omega=\\{V_{1},\dots,V_{t}\\}$ and fix an $F$-orbit
$\Omega_{1}$ in $\Omega$. Throughout, $W=\Sigma_{U\in\Omega_{1}}U.$

Now, we split the proof into a sequence of steps.

(1) We may assume that $Q$ acts faithfully on $V$. Furthermore $Ker(C_{Q}(H)\
\textrm{on}\ C_{V}(H))=Ker(C_{Q}(H)\ \textrm{on}\ V)=1$.

###### Proof.

Suppose that $Ker(Q\ \rm{on}\ V)\neq 1$ and set $\overline{Q}=Q/Ker(Q\
\rm{on}\ V)$. Note that since $C_{Q}(F)=1$, $F$ is a Carter subgroup of $QF$
and hence also a Carter subgroup of $\overline{Q}F$ which implies that
$C_{\overline{Q}}(F)=1$. Notice that the equality
$\overline{C_{Q}(H)}=C_{\overline{Q}}(H)$ holds in case $FH$ is Frobenius (see
[14] Theorem 2.3). The same equality holds in case where $FH$ is non-Frobenius
due to the coprimeness condition $(q,|H|)=1.$ Then
$[C_{V}(H),C_{Q}(H)]=[C_{V}(H),C_{\overline{Q}}(H)]$ and so we may assume that
$Q$ acts faithfully on $V$. Notice that by Proposition 2.3 we have

$Ker(C_{Q}(H)\ \textrm{on}\ C_{V}(H))=Ker(C_{Q}(H)\ \textrm{on}\ V)=1$

establishing the claim. ∎

(2) We may assume that $Q=\langle c^{F}\rangle$ for any nonidentity element
$c\in C_{Z(Q)}(H)$ of order $q$. In particular $Q$ is abelian.

###### Proof.

We obtain that $C_{Z(Q)}(H)\neq 1$ as $C_{Q}(F)=1$ by Proposition 2.3. Let now
$1\neq c\in C_{Z(Q)}(H)$ of order $q$ and consider $\langle
c^{FH}\rangle=\langle c^{F}\rangle$, the minimal $FH$-invariant subgroup
containing $c$. Since $V$ is an irreducible $QFH$-module on which $Q$ acts
faithfully we have that $V=[V,\langle c^{F}\rangle]$. Thus we may assume that
$Q=\langle c^{F}\rangle$ as claimed. ∎

(3) $V=[V,c]\cdot[V,c^{f_{1}}]\cdots[V,c^{f_{n}}]$ where $n$ is a
$(s,|H|)$-bounded number. Hence it suffices to bound $\mathbf{r}([W,c])$.

###### Proof.

Notice that the group $C_{Q}(H)$ embeds in the automorphism group of
$[C_{V}(H),C_{Q}(H)]$ by step $\it(1)$. Then $C_{Q}(H)$ has $s$-bounded rank
by Lemma 2.7. This yields by Theorem 2.2 that $Q$ has $(s,|H|)$-bounded rank.
Thus, there exist $f_{1}=1,\ldots,f_{n}$ in $F$ for an $(s,|H|)$-bounded
number $n$ such that $Q=\langle c^{f_{1}},\ldots,c^{f_{n}}\rangle$. Now
$V=[V,c]\cdot[V,c^{f_{2}}]\cdots[V,c^{f_{n}}]=\prod_{i=1}^{n}[V,c]^{f_{i}}$ by
Lemma 2.6. This shows that we need only to bound $\mathbf{r}([V,c])$ suitably.
In fact it suffices to show that $\mathbf{r}([W,c])$ is suitably bounded as
$V=\Sigma_{h\in H}W^{h}.$ ∎

(4) $H_{1}=Stab_{H}(\Omega_{1})\neq 1$. Furthermore the rank of the sum of
members of $\Omega_{1}$ which are not centralized by $c$ and contained in a
regular $H_{1}$-orbit, is suitably bounded.

###### Proof.

Fix $U\in\Omega_{1}$ and set $Stab_{F}(U)=F_{1}$. Choose a transversal $T$ for
$F_{1}$ in $F.$ Let $W=\sum_{t\in T}U^{t}$ where $T$ is a transversal for
$F_{1}$ in $F$ with $1\in T.$ Then we have $V=\sum_{h\in H}W^{h}$. Notice that
$[V,c]\neq 0$ by $\it(1)$ which implies that $[W,c]\neq 0$ and hence
$[U^{t},c]=U^{t}$ for some $t\in T$. Without loss of generality we may assume
that $[U,c]=U.$

Suppose that $Stab_{H}(\Omega_{1})=1$. Then we also have $Stab_{H}(U^{t})=1$
for all $t\in T$ and hence the sum $X_{t}=\sum_{h\in H}U^{th}$ is direct for
all $t\in T.$ Now, $U\leq X_{1}$. It holds that

$C_{X_{t}}(H)=\\{\sum_{h\in H}v^{h}\ :\ v\in U^{t}\\}.$

Then $|U|=|C_{X_{1}}(H)|=|[C_{X_{1}}(H),c]|\leq|[C_{V}(H),C_{Q}(H)]|$ implies
$\mathbf{r}(U)\leq s.$ On the other hand $V=\bigoplus_{t\in T}X_{t}$ and

$[C_{V}(H),c]=\bigoplus\\{[C_{X_{t}}(H),c]:t\in T\,\,\text{with
}\,\,[U^{t},c]\neq 0\\}\leq[C_{V}(H),C_{Q}(H)].$

In particular, $\\{t\in T:[U^{t},c]\neq 0\\}$ is suitably bounded whence
$\mathbf{r}([W,c])$ is $(s,|H|)$-bounded. Hence we may assume that
$Stab_{H}(\Omega_{1})\neq 1.$

Notice that every element of a regular $H_{1}$-orbit in $\Omega_{1}$ lies in a
regular $H$-orbit in $\Omega$. Let $U\in\Omega_{1}$ be contained in a regular
$H_{1}$-orbit of $\Omega_{1}.$ Let $X$ denote the sum of the members of the
$H$-orbit of $U$ in $\Omega$, that is $X=\bigoplus_{h\in H}U^{h}$. Then
$C_{X}(H)=\\{\sum_{h\in H}v^{h}\ :\ v\in U\\}$. If $[U,c]\neq 0$ then by
repeating the same argument in the above paragraph we show that
$\mathbf{r}(U)\leq s$ is suitably bounded. On the other hand the number, say
$m$, of all $H$-orbits in $\Omega$ containing a member $U$ such that
$[U,c]\neq 0$ is suitably bounded because $m\leq\mathbf{r}([C_{V}(H),c])\leq
s.$ It follows then that the rank of the sum of members of $\Omega_{1}$ which
are not centralized by $c$ and contained in a regular $H_{1}$-orbit, is
suitably bounded. ∎

(5) We may assume that $FH$ is not Frobenius.

###### Proof.

Assume the contrary that $FH$ is Frobenius. Let $H_{1}=Stab_{H}(\Omega_{1})$
and pick $U\in\Omega_{1}$. Set $S=Stab_{FH_{1}}(U)$ and $F_{1}=F\cap S$. Then
$|F:F_{1}|=|\Omega_{1}|=|FH_{1}:S|$ and so $|S:F_{1}|=|H_{1}|$. Since
$(|F_{1}|,|H_{1}|)=1$, by the Schur-Zassenhaus theorem there exists a
complement, say $S_{1}$ of $F_{1}$ in $S$ with $|H_{1}|=|S_{1}|$. Therefore
there exists a conjugate of $U$ which is $H_{1}$-invariant. There is no loss
in assuming that $U$ is $H_{1}$-invariant. On the other hand if $1\neq h\in
H_{1}$ and $x\in F$ such that $U^{xh}=U^{x}$, then $[h,x]\in
Stab_{F}(U)=F_{1}$ and so $F_{1}x=F_{1}x^{h}=(F_{1}x)^{h}$. This implies that
$F_{1}x\cap C_{F}(h)$ is nonempty. Now the Frobenius action of $H$ on $F$
forces that $x\in F_{1}$. This means that for each $x\in F\setminus F_{1}$ we
have $Stab_{H_{1}}(U^{x})=1$. Therefore $U$ is the unique member of
$\Omega_{1}$ which is $H_{1}$-invariant and all the $H_{1}$-orbits other than
$\\{U\\}$ are regular. By $\it(4)$, the rank of the sum of all members of
$\Omega_{1}$ other than $U$ is is suitably bounded. In particular
$\mathbf{r}(U)$ and hence $\mathbf{r}([W,c])$ is suitably bounded in case
where $[U^{x},c]\neq 0$ for some $x\in F\setminus F_{1}$. Thus we may assume
that $c$ is trivial on $U^{x}$ for all $x\in F\setminus F_{1}$. Now we have
$[W,c]=[U,c]=U.$

Due to the action by scalars of the abelian group $Q$ on $U$, it holds that
$[Q,F_{1}]\leq C_{Q}(U)$. We also know that $c^{x}$ is trivial on $U$ for each
$x\in F\setminus F_{1}$. Since $C_{Q}(F)=1$, there are prime divisors of $|F|$
different from $q.$ Let $F_{q^{\prime}}$ denote the $q^{\prime}$-Hall subgroup
of $F.$ Clearly we have $C_{Q}(F_{q^{\prime}})=1$. Let now $y=\prod_{f\in
F_{q^{\prime}}}c^{f}$. Then we have

$1=y=(\prod_{f\in F_{1}\cap F_{q^{\prime}}}c^{f})(\prod_{f\in
F_{q^{\prime}}\setminus F_{1}}c^{f})\in c^{|F_{1}\cap
F_{q^{\prime}}|}C_{Q}(U).$

As a consequence $c\in C_{Q}(U)$, because $q$ is coprime to
$|F_{q^{\prime}}|$. This contradiction establishes the claim. ∎

(6) We may assume that the group $FH$ is Frobenius-like of Type II.

###### Proof.

On the contrary we assume that $FH$ is Frobenius-like of Type I. By $\it(4),$
we have $H_{1}=Stab_{H}(\Omega_{1})\neq 1$. Choose a transversal $T_{1}$ for
$H_{1}$ in $H.$ Now $V=\bigoplus_{h\in T_{1}}W^{h}.$ Also we can guarantee the
existence of a conjugate of $U$ which is $H_{1}$-invariant by means of the
Schur-Zassenhaus Theorem as in $\it(5)$. There is no loss in assuming that $U$
is $H_{1}$-invariant.

Set now $Y=\Sigma_{x\in F^{\prime}}U^{x}$ and $F_{2}=Stab_{F}(Y)$ and
$F_{1}=Stab_{F}(U).$ Clearly, $F_{2}=F^{\prime}F_{1}$ and $Y$ is
$H_{1}$-invariant. Notice that for all nonidentity $h\in H$, we have
$C_{F}(h)\leq F^{\prime}\leq F_{2}$ . Assume first that $F=F_{2}$. This forces
that we have $V=Y$. Clearly, $Y\neq U$, that is $F^{\prime}\not\leq F_{1}$,
because otherwise $Q=[Q,F]=1$ due to the scalar action of the abelian group
$Q$ on $U$. So $F^{\prime}\cap F_{1}=1$ which implies that $|F:F_{1}|$ is a
prime. Then $F_{1}\unlhd F$ and $F^{\prime}\leq F_{1}$ which is impossible.
Therefore $F\neq F_{2}$.

If $1\neq h\in H$ and $t\in F$ such that ${Y}^{th}={Y}^{t}$ then $[h,t]\in
F_{2}$. Now, $F_{2}t=F_{2}t^{h}=(F_{2}t)^{h}$ and this implies the existence
of an element in $F_{2}t\cap C_{F}(h)$. Since $C_{F}(h)\leq F^{\prime}\leq
F_{2}$ we get $t\in F_{2}$. In particular, for each $t\in F\setminus F_{2}$ we
have $Stab_{H}(Y^{t})=1$.

Let $S$ be a transversal for $F_{2}$ in $F$. For any $t\in S\setminus F_{2}$
set $Y_{t}=Y^{t}$ and consider $Z_{t}=\Sigma_{h\in H}{Y_{t}}^{h}$. Notice that
$V=Y\oplus\bigoplus_{t\in S\setminus F_{2}}Z_{t}$. As the sum $Z_{t}$ is
direct we have

$C_{Z_{t}}(H)=\\{\sum_{h\in H}v^{h}\ :\ v\in Y_{t}\\}$

with $|C_{Z_{t}}(H)|=|Y_{t}|.$ Then
$\mathbf{r}([Y_{t},c])=\mathbf{r}([C_{Z_{t}}(H),c])\leq s$ for each $t\in
S\setminus F_{2}$ with $[Y_{t},c]\neq 0$. On the other hand,

$\Sigma\\{\mathbf{r}([C_{Z_{t}}(H),c]):t\in S\,\,\text{with}\,\,[Y_{t},c]\neq
0\\}\leq\mathbf{r}([C_{V}(H),c])\leq s$

whence $|\\{t\in S\setminus F_{2}:[Y_{t},c]\neq 0\\}|$ is suitably bounded. So
the claim is established if there exists $t\in S\setminus F_{2}$ such that
$[Y_{t},c]\neq 0$, since we have $V=Y\oplus\bigoplus_{t\in S\setminus
F_{2}}Z_{t}$. Thus we may assume that $c$ is trivial on $\bigoplus_{t\in
S\setminus F_{2}}Z_{t}$ and hence $[V,c]=[Y,c].$

There are two cases now: We have either $F^{\prime}\cap F_{1}=1$ or
$F^{\prime}\leq F_{1}.$ First assume that $F^{\prime}\leq F_{1}.$ Then we get
$F_{1}=F_{2}$ because $F_{2}=F^{\prime}F_{1}.$ Now $U=Y.$ Due to the action by
scalars of the abelian group $Q$ on $U$, it holds that $[Q,F_{1}]\leq
C_{Q}(U)$. From this point on we can proceed as in the proof of step $\it(5)$
and observe that $C_{Q}(F_{q^{\prime}})=1$. Letting now $y=\prod_{f\in
F_{q^{\prime}}}c^{f}$, we have

$1=y=(\prod_{f\in F_{1}\cap F_{q^{\prime}}}c^{f})(\prod_{f\in
F_{q^{\prime}}\setminus F_{1}}c^{f})\in c^{|F_{1}\cap
F_{q^{\prime}}|}C_{Q}(U).$

implying that $c\in C_{Q}(U)$, because $q$ is coprime to $|F_{q^{\prime}}|$.

Thus we have $F_{1}\cap F^{\prime}=1$. First assume that $H_{1}=H$. Then $Y$
is $H$-invariant and $F_{1}H$ is a Frobenius group. Note that $C_{U}(F_{1})=1$
as $C_{V}(F)=1$, and hence $C_{Y}(F_{1})=1$ since $F^{\prime}\leq Z(F).$ We
consider now the action of $QF_{1}H$ on $Y$ and the fact that
$\mathbf{r}([C_{Y}(H),C_{Q}(H)])\leq s.$ Then step $\it(5)$, we obtain that
$\mathbf{r}(Y)=\mathbf{r}([Y,Q])$ is $(s,|H|)$-bounded. Next assume that
$H_{1}\neq H.$ Choose a transversal for $H_{1}$ in $H$ and set
$Y_{1}=\Sigma_{h\in T_{1}}Y^{h}$. Clearly this sum is direct and hence

$C_{Y_{1}}(H)=\\{\sum_{h\in T_{1}}v^{h}\ :\ v\in Y\\}$

with $|[C_{Y_{1}}(H),c]|=|[Y,c]|.$ Then
$\mathbf{r}([Y,c])=\mathbf{r}([C_{Y_{1}}(H),c])\leq s$ establishing claim
$\it(6)$. ∎

(7) The proposition follows.

###### Proof.

From now on $FH$ is a Frobenius-like group of Type II, that is, $H$ and
$C_{F}(H)$ are of prime orders. By step $\it(4)$ we have
$H=H_{1}=Stab_{H}(\Omega_{1})$ since $|H|$ is a prime. Now $V=W$. We may also
assume by the Schur-Zassenhaus theorem as in the previous steps that there is
an $H$-invariant element, say $U$ in $\Omega$. Let $T$ be a transversal for
$F_{1}=Stab_{F}(U)$ in $F$. Then $F=\bigcup_{t\in T}{F_{1}}t$ implies
$V=\bigoplus_{t\in T}U^{t}$. It should also be noted that we have $|\\{t\in
T:[U^{t},c]\neq 0\\}|$ is suitably bounded as

$[C_{V}(H),c]=\bigoplus\\{[C_{X_{t}}(H),c]:t\in T\,\,\text{with
}\,\,[U^{t},c]\neq 0\\}\leq[C_{V}(H),C_{Q}(H)]$

where $X_{t}=\bigoplus_{h\in H}U^{th}$.

Let $X$ be the sum of the components of all regular $H$-orbits on $\Omega$,
and let $Y$ denote the sum of all $H$-invariant elements of $\Omega$. Then
$V=X\oplus Y.$ Suppose that ${U}^{th}={U}^{t}$ for $t\in T$ and $1\neq h\in
H$. Now $[t,h]\in F_{1}$ and so the coset $F_{1}t$ is fixed by $H$. Since the
orders of $F$ and $H$ are relatively prime we may assume that $t\in C_{F}(H).$
Conversely for each $t\in C_{F}(H)$, ${U}^{t}$ is $H$-invariant. Hence the
number of components in $Y$ is $|T\cap C_{F}(H)|=|C_{F}(H):C_{F_{1}}(H)|$ and
so we have either $C_{F}(H)\leq F_{1}$ or not.

If $C_{F}(H)\not\leq F_{1}$ then $C_{F_{1}}(H)=1$ whence $F_{1}H$ is Frobenius
group acting on $U$ in such a way that $C_{U}(F_{1})=1$. Then $\mathbf{r}(U)$
is $(s,|H|)$-bounded by step $\it(5)$ since
$\mathbf{r}([C_{U}(H),C_{Q}(H)])\leq s$ holds. This forces that
$\mathbf{r}([V,c])$ is bounded suitably and hence the claim is established.

Thus we may assume that $C_{F}(H)\leq F_{1}.$ Then $Y=U$ is the unique
$H$-invariant $Q$-homogeneous component. If $[U^{t},c]\neq 0$ for some $t\in
F\setminus F_{1}$ we can bound $\mathbf{r}(U)$ and hence $\mathbf{r}([V,c])$
suitably. Thus we may assume that $c$ is trivial on $U^{t}$ for each $t\in
F\setminus F_{1}.$ Due to the action of the abelian group $Q$ on $U$, it holds
that $[Q,F_{1}]\leq C_{Q}(U)$. From this point on we can proceed as in the
proof of step $\it(5)$ and observe that $C_{Q}(F_{q^{\prime}})=1$. Letting now
$y=\prod_{f\in F_{q^{\prime}}}c^{f}$, we have

$1=y=(\prod_{f\in F_{1}\cap F_{q^{\prime}}}c^{f})(\prod_{f\in
F_{q^{\prime}}\setminus F_{1}}c^{f})\in c^{|F_{1}\cap
F_{q^{\prime}}|}C_{Q}(U).$

implying that $c\in C_{Q}(U)$, because $q$ is coprime to $|F_{q^{\prime}}|$.
This final contradiction completes the proof of Proposition 3.1. ∎

∎

The next proposition studies the action of a dihedral group of automorphisms
and is essential in proving Theorem B.

######  Proposition 3.2.

Let $D=\langle\alpha,\beta\rangle$ be a dihedral group generated by two
involutions $\alpha$ and $\beta$. Suppose that $D$ acts on a $q$-group $Q$ for
some prime $q$. Let $V$ be an irreducible $\mathbb{F}_{p}QD$-module where
$\mathbb{F}_{p}$ is a field with characteristic $p$ not dividing $|Q|$.
Suppose that $C_{VQ}(F)=1$ where $F=\langle\alpha\beta\rangle$. If
$max\\{\mathbf{r}([C_{V}(\alpha),C_{Q}(\alpha)]),\mathbf{r}([C_{V}(\beta),C_{Q}(\beta)])\\}\leq
s$, then $\mathbf{r}([V,Q])$ is $s$-bounded.

###### Proof.

We set $H=\langle\alpha\rangle$. So $D=FH$. By Lemma 2.6 and Theorem 2.4, we
have $[V,Q]=[V,C_{Q}(\alpha)][V,C_{Q}(\beta)]$. Then it is sufficient to bound
the rank of $[V,C_{Q}(H)]$. Following the same steps as in the proof of
Proposition 3.1 by replacing Proposition 2.3 by Proposition 2.4, we observe
that $Q$ acts faithfully on $V$ and $Q=\langle c^{F}\rangle$ is abelian with
$c\in C_{Z(Q)}(H)$ of order $q$. Furthermore $Ker(C_{Q}(H)\ \textrm{on}\
C_{V}(H))=Ker(C_{Q}(H)\ \textrm{on}\ V)=1$. Note that it suffices to bound
$\mathbf{r}([V,c])$ suitably.

Let $\Omega$ denote the set of $Q$-homogeneous components of the irreducible
$QD$-module $V.$ Let $\Omega_{1}$ be an $F$-orbit of $\Omega$ and set
$W=\Sigma_{U\in\Omega_{1}}U.$ Then we have $V=W+W^{\alpha}$. Suppose that
$W^{\alpha}\neq W$. Then for any $U\in\Omega_{1}$ we have $Stab_{H}(U)=1$. Let
$T$ be a tranversal for $Stab_{F}(U)=F_{1}$ in $F$ . It holds that
$V=\Sigma_{t\in T}X_{t}$ where $X_{t}=U^{t}+U^{t\alpha}.$ Now
$[V,c]=\Sigma_{t\in T}[X_{t},c]$ and $C_{V}(H)=\Sigma_{t\in T}C_{X_{t}}(H)$
where $C_{X_{t}}(H)=\\{w+w^{\alpha}:w\in U^{t}\\}$. Since $[V,c]\neq 0$ there
exists $t\in T$ such that $[U^{t},c]\neq 0$, that is $[U^{t},c]=U^{t}.$ Then
$[C_{X_{t}}(H),c]=C_{X_{t}}(H).$ Since $\mathbf{r}([C_{V}(H),C_{Q}(H)])\leq s$
we get $\mathbf{r}(U)=\mathbf{r}(C_{X_{t}}(H))\leq s$. Furthermore it follows
that $|\\{t\in T:[U^{t},c]\neq 0\\}|$ is $s$-bounded and as a consequence
$\mathbf{r}([V,c])$ is suitably bounded. Thus we may assume that
$W^{\alpha}=W$ which implies that $\Omega_{1}=\Omega$ and $H$ fixes an
element, say $U$, of $\Omega$ as desired.

Let $U^{t}\in\Omega$ be $H$-invariant. Then $[t,\alpha]\in F_{1}.$ On the
other hand $t^{-1}t^{\alpha}=t^{-2}$ since $\alpha$ inverts $F$. So $F_{1}t$
is an element of $F/F_{1}$ of order at most $2$ which implies that the number
of $H$-invariant elements of $\Omega$ is at most $2$. Let now $Y$ be the sum
of all $H$-invariant elements of $\Omega$. Then
$V=Y\oplus\bigoplus_{i=1}^{m}X_{i}$ where $X_{1},\ldots X_{m}$ are the sums of
elements in $H$-orbits of length $2.$ Let $X_{i}=U_{i}\oplus U_{i}^{\alpha}$.
Notice that if $[U_{i},c]\neq 0$ for some $i$, then we obtain
$\mathbf{r}(U)=\mathbf{r}(U_{i})\leq s$ by a similar argument as above. On the
other hand we observe that the number of $i$ for which $[U_{i},c]\neq 0$ is
$s$-bounded by the the hypothesis that $\mathbf{r}([C_{V}(H),c])\leq s$. It
follows now that $\mathbf{r}([V,c])$ is suitably bounded in case where
$[U_{i},c]\neq 0$ for some $i$.

Thus we may assume that $c$ centralizes $\bigoplus_{i=1}^{m}X_{i}$ and that
$[U,c]=U$. Due to the scalar action by scalars of the abelian group $Q$ on
$U$, it holds that $[Q,F_{1}]\leq C_{Q}(U)$. As $F_{1}\unlhd FH$, we have
$[Q,F_{1}]\leq C_{Q}(V)=1$. Clearly we have $C_{Q}(F_{q^{\prime}})=1$ where
$F_{q^{\prime}}$ denotes the Hall $q^{\prime}$-part of $F$ whose existence is
guaranteed by the fact that $C_{Q}(F)=1.$ Let now $y=\prod_{f\in
F_{q^{\prime}}}c^{f}$. Then we have

$1=y=(\prod_{f\in F_{1}\cap F_{q^{\prime}}}c^{f})(\prod_{f\in
F_{q^{\prime}}\setminus F_{1}}c^{f})\in c^{|F_{1}\cap
F_{q^{\prime}}|}C_{Q}(U).$

As a consequence $c\in C_{Q}(U)$, because $q$ is coprime to
$|F_{q^{\prime}}|$. This contradiction completes the proof of Proposition 3.2.

∎

##  4\. Proofs of theorems

Firstly, we shall give a detailed proof for Theorem A part (b). The proof of
Theorem A (a) can be easily obtained by just obvious modifications of the
proof of part (b).

First, we assume that $G=PQ$ where $P$ and $Q$ are $FH$-invariant subgroups
such that $P$ is a normal $p$-subgroup for a prime $p$ and $Q$ is a nilpotent
$p^{\prime}$-group with $|[C_{P}(H),C_{Q}(H)]|=p^{s}$. We shall prove that
$\mathbf{r}(\gamma_{\infty}(G))$ is $((s,|H|)$-bounded. Clearly
$\gamma_{\infty}(G)=[P,Q]$. Consider an unrefinable $FH$-invariant normal
series

$P=P_{1}>P_{2}>\cdots>P_{k}>P_{k+1}=1.$

Note that its factors $P_{i}/P_{i+1}$ are elementary abelian. Let $V=P_{k}$.
Since $C_{V}(Q)=1$, we have that $V=[V,Q]$. We can also assume that $Q$ acts
faithfully on $V$. Proposition 3.1 yields that $\mathbf{r}(V)$ is
$(s,|H|)$-bounded. Set $S_{i}=P_{i}/P_{i+1}$. If $[C_{S_{i}}(H),C_{Q}(H)]=1$,
then $[S_{i},Q]=1$ by Proposition 2.3. Since $C_{P}(Q)=1$ we conclude that
each factor $S_{i}$ contains a nontrivial image of an element of
$[C_{P}(H),C_{Q}(H)]$. This forces that $k\leq s$. Then we proceed by
induction on $k$ to obtain that $\mathbf{r}([P,Q])$ is an $(s,|H|)$-bounded
number, as desired.

Let $F(G)$ denote the Fitting subgroup of a group $G$. Write $F_{0}(G)=1$ and
let $F_{i+1}(G)$ be the inverse image of $F(G/F_{i}(G))$. As is well known,
when $G$ is soluble, the least number $h$ such that $F_{h}(G)=G$ is called the
Fitting height $h(G)$ of $G$. Let now $r$ be the rank of
$\gamma_{\infty}(C_{G}(H))$. Then $C_{G}(H)$ has $r$-bounded Fitting height
(see for example Lemma 1.4 of [15]) and hence $G$ has $(r,|H|)$-bounded
Fitting height.

We shall proceed by induction on $h(G)$. Firstly, we consider the case where
$h(G)=2$. Indeed, let $P$ be a Sylow $p$-subgroup of $\gamma_{\infty}(G)$ and
$Q$ an $FH$-invariant Hall $p^{\prime}$-subgroup of $G$. Then, by the
preceeding paragraphs and Lemma 2.8, the rank of $P=[P,Q]$ is
$(r,|H|)$-bounded and so the rank of $\gamma_{\infty}(G)$ is
$(r,|H|)$-bounded. Assume next that $h(G)>2$ and let $N=F_{2}(G)$ be the
second term of the Fitting series of $G$. It is clear that the Fitting height
of $G/\gamma_{\infty}(N)$ is $h-1$ and
$\gamma_{\infty}(N)\leq\gamma_{\infty}(G)$. Hence, by induction we have that
$\gamma_{\infty}(G)/\gamma_{\infty}(N)$ has $(r,|H|)$-bounded rank. As a
consequence, it holds that

${\bf r}(\gamma_{\infty}(G))\leq{\bf
r}(\gamma_{\infty}(G)/\gamma_{\infty}(N))+{\bf r}(\gamma_{\infty}(N))$

completing the proof of Theorem A(b).

The proof of Theorem B can be directly obtained as in the above argument by
replacing Proposition 3.1 by Proposition 3.2; and Proposition 2.3 by
Proposition 2.5.

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