# Bounds on the number of maximal subgroups of finite groups

A. Ballester-Bolinches Department of Mathematics, Guandong University of
Education, Guangzhou, Guangdong, 510303, People’s Republic of ChinaDepartament
de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100 Burjassot,
València, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>ORCID
0000-0002-2051-9075, 0000-0002-2321-8139 R. Esteban-Romero00footnotemark: 0 P.
Jiménez-Seral Departamento de Matemáticas, Universidad de Zaragoza, Pedro
Cerbuna, 12, 50009 Zaragoza, Spain<EMAIL_ADDRESS>ORCID 0000-0003-4809-1784

###### Abstract

In this paper we obtain significant bounds for the number of maximal subgroups
of a given index of a finite group. These results allow us to give new bounds
for the number of random generators needed to generate a finite $d$-generated
group with high probability.

_Keywords:_ finite group, maximal subgroup, probabilistic generation,
primitive group

_Mathematics Subject Clasification (2020):_ 20P05, 20E07, 20E28

##  1 Introduction

All groups considered in this paper will be finite.

The fact that an $r$-tuple of elements of a group $G$ does not generate $G$ is
equivalent to the fact that all these elements belong to a certain maximal
subgroup $M$ of $G$. The probability that a certain $r$-tuple of elements of
$G$ chosen at random and with a uniform probability distribution is contained
in a maximal subgroup $M$ of $G$ is $1/\lvert G:M\rvert^{r}$. This makes
interesting to consider the number $\operatorname{m}_{n}(G)$ of maximal
subgroups of a group $G$ of index $n$ for a natural number $n$ in problems
related with the probability that a certain tuple of elements of the group
generates it.

Pak [Pak], motivated by potential applications for the product replacement
algorithm, widely used to generate random elements in a finitely generated
group, analysed an invariant related to the probability a tuple generates a
group.

######  Definition 1.1.

Given a group $G$, we denote by $\mathcal{V}(G)$ the least positive integer
$k$ such that $G$ is generated by $k$ random elements with probability at
least $1/\mathrm{e}$.

Pak conjectured that
$\mathcal{V}(G)={\operatorname{O}(\operatorname{d}(G)\log\log\lvert
G\rvert)}$, where $\operatorname{d}(G)$ denotes the smallest cardinal of a
generating set for $G$. Here and in all this paper, the symbol $\log$ will be
used to denote the logarithm to the base $2$, and we follow the convention
that $\log 0=-\infty$.

Given a maximal subgroup $M$ of a group $G$, the quotient $G/M_{G}$ of $G$ by
the normal core $M_{G}$ of $M$ in $G$ is a primitive group. According to a
well-known theorem of Baer [Bae57] (see also [BBE06, Theorem 1.1.7]),
$G/M_{G}$ is of one of three different types. In this case, we say that the
maximal $M$ is of the same type as the primitive group $G/M_{G}$ it induces,
like in [BBE06, Definition 1.1.19]. Lubotzky [Lub02] established bounds on the
number of maximal subgroups of type 1 of a given index $n$ and on the number
of maximal subgroups of types 2 and 3 of index $n$ by considering the number
of abelian and non-abelian chief factors, respectively, in a given chief
series of the group. He obtained the following result that confirms the
validity of Pak’s conjecture.

######  Theorem 1.2 (Lubotzky, [Lub02, Corollary 2.6]).

If $G$ is a group with $r$ chief factors in a given chief series, $r_{a}$ of
them abelian and $r_{b}$ of them non-abelian, then

$\operatorname{m}_{n}(G)\leq\bigl{(}\tfrac{1}{2}(r_{b}+1)r_{b}+r_{a}n^{\operatorname{d}(G)}\bigl{)}n^{2}\leq
r^{2}n^{\operatorname{d}(G)+2}.$

Furthermore,

$\mathcal{V}(G)\leq\frac{1+\log\log\lvert
G\rvert}{\log{\operatorname{i}(G)}}+\max\left(\operatorname{d}(G),\frac{\log\log\lvert
G\rvert}{\log{\operatorname{i}(G)}}\right)+2.02,$

where $\operatorname{i}(G)$ denotes the smallest index of a proper subgroup of
$G$.

The bound for $\mathcal{V}(G)$ in Theorem 1.2 depends on the following
invariant.

######  Definition 1.3.

For a group $G$, let us call

$\mathcal{M}(G)=\max_{n\geq 2}\log_{n}\operatorname{m}_{n}(G)=\max_{n\geq
2}\frac{\log\operatorname{m}_{n}(G)}{\log n}.$

Here we use $\log_{n}x$ to denote the logarithm to the base $n$ of $x$, that
is, $\log_{n}x=\log x/\log n=\ln x/\ln n$. There is a close relation between
$\mathcal{M}(G)$ and $\mathcal{V}(G)$.

######  Theorem 1.4 (Lubotzky, [Lub02, Proposition 1.2]).

$\mathcal{M}(G)-3.5\leq\mathcal{V}(G)\leq\mathcal{M}(G)+2.02.$

The proof of the left hand side bound in the above result depends on the
following result.

######  Theorem 1.5 (see [Lub02, Theorem 1.3] or [DL06, Theorem 21]).

There exists a constant $b$ such that for every group $G$ and every $n\geq 2$,
$G$ has at most $n^{b}$ core-free maximal subgroups of index $n$. In fact,
$b=2$ will do.

Unless otherwise stated, we will follow the notation of the books [DH92] and
[BBE06]. Detailed information about primitive groups and chief factors,
crowns, and precrowns of a group can be found in [BBE06, Chapter 1].

Detomi and Lucchini [DL03], with the help of the number $\lambda(G)$ of non-
Frattini chief factors in a given chief series of $G$ and by considering their
associate crowns, proved independently also the validity of Pak’s conjecture.

######  Theorem 1.6 (Detomi and Lucchini [DL03, Theorem 20]).

There exists a constant $c$ such that, for any group $G$,
$\mathcal{V}(G)\leq\lfloor\operatorname{d}(G)+c\log\lambda(G)\rfloor$ if
$\lambda(G)>1$, otherwise,
$\mathcal{V}(G)\leq\lfloor\operatorname{d}(G)+c\rfloor$, where $\lambda(G)$
denotes the number of non-Frattini chief factors in a given chief series of
$G$.

Here the symbol $\lfloor x\rfloor$ denotes the defect integer part of $x$,
that is, the largest integer number $n$ such that $n\leq x$.

The proof of this result uses some results of Dalla Volta and Lucchini [DVL98]
and Dalla Volta, Lucchini, and Morini [DVLM01] about the number of generators
of powers of certain diagonal-type subgroups of direct powers of copies of a
monolithic group.

The aim of this paper is to obtain tighter bounds for
$\operatorname{m}_{n}(G)$, and so for $\mathcal{V}(G)$, by considering the
numbers of maximal subgroups of each type, as in Lubotzky’s paper [Lub02], and
with the help of the crowns associated to abelian chief factors, like in
Detomi and Lucchini’s paper [DL03], and to non-abelian chief factors. Our
bounds will depend on four invariants. Our first invariant is related to
maximal subgroups of type 1.

######  Definition 1.7.

Let $G$ be a group and let $n>1$ be a natural number. We denote by
$\operatorname{cr}^{\mathfrak{A}}_{n}(G)$ the number of crowns associated to
complemented abelian chief factors of order $n$ of $G$, that is, the number of
$G$-isomorphism classes of complemented abelian chief factors of $G$.

Clearly, $\operatorname{cr}^{\mathfrak{A}}_{n}(G)=0$ unless $n$ is a power of
a prime.

In order to define our second invariant, which concerns non-abelian chief
factors and is related to the primitive quotients of type 2, we must recall
the next definition.

######  Definition 1.8 (see [BBE06, Definition 1.2.9]).

Given a chief factor $H/K$ of a group $G$, the _primitive group $[H/K]*G$
associated with $H/K$ in $G$_ coincides with the semidirect product
$[H/K]\bigl{(}G/{\operatorname{C}_{G}(H/K)}\bigr{)}$ if $H/K$ is abelian and
with the quotient group $G/{\operatorname{C}_{G}(H/K)}$ if $H/K$ is non-
abelian.

######  Definition 1.9.

Let $n$ be a natural number. The symbol ${\operatorname{rs}_{n}(G)}$ denotes
the number of non-abelian chief factors $A$ in a given chief series of $G$
such that the associated primitive group $[A]*G$ has a core-free maximal
subgroup of index $n$.

Our third and fourth invariants concern also non-abelian chief factors and
contain information about the primitive quotients of type 3 of the group.

######  Definition 1.10.

Let $n$ be a natural number. The symbol ${\operatorname{ro}_{n}(G)}$ denotes
the number of non-abelian chief factors $A$ in a given chief series of $G$
such that $A$ has order $n$.

For the definition of our fourth invariant, the following auxiliary invariant
is useful.

######  Definition 1.11.

Let $G$ be a group. For a characteristically simple group $A$, that is, a
direct product of copies of a simple group $S$, $\operatorname{r}_{A}(G)$
denotes the number of non-Frattini chief factors isomorphic to $A$ in a given
chief series of $G$.

According to Theorem A of [BBERJS], $\operatorname{r}_{A}(G)$ coincides with
the largest number $r$ such that $G$ has a normal section that is the direct
product of $r$ non-Frattini chief factors of $G$ that are isomorphic (not
necessarily $G$-isomorphic) to $A$.

######  Definition 1.12.

Let $n$ be a natural number. The symbol ${\operatorname{rm}_{n}(G)}$ denotes
the maximum of the lenghts of the $G$-crowns associated to non-abelian chief
factors of order $n$ of $G$.

We will see that, according to Theorem 2.6, the non-abelian chief factors $A$
of $G$ of order $n$ fall into at most two isomorphism classes. We also note
that a non-abelian chief factor can be counted in $\operatorname{rs}_{n}(G)$
for different values of $n$, if the corresponding primitive group has core-
free maximal subgroups of different indices. Furthermore, a non-abelian chief
factor of order $n$ that is counted in $\operatorname{ro}_{n}(G)$ can also be
counted in $\operatorname{rs}_{n}(G)$ if the primitive group associated to the
corresponding chief factor has a maximal subgroup complementing its socle (see
[JSL98] or [BBE06, Theorem 1.1.48]),

We obtain a bound for $\operatorname{m}_{n}(G)$ that improves the one of
Theorem 1.2.

######  Theorem A.

Let $G$ be a non-cyclic group with $r$ chief factors in a given chief series.
For every natural $n\geq 2$, $\operatorname{m}_{n}(G)\leq
rn^{\operatorname{d}(G)+2}$.

This bound will appear as a consequence of a tighter bound in terms of the
invariants $\operatorname{cr}^{\mathfrak{A}}_{n}(G)$,
$\operatorname{rs}_{n}(G)$, $\operatorname{rm}_{n}(G)$, and
$\operatorname{ro}_{n}(G)$ presented in Theorem 2.10. We also obtain a bound
for $\mathcal{V}(G)$.

######  Theorem B.

Let $G$ be a $d$-generated non-trivial group with $d\geq 2$. Then

$\mathcal{V}(G)\leq\eta(G),$

where $\eta(G)$ is a function bounded by a linear combination of $d$ and the
maxima of $\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G)$,
$\log_{n}\operatorname{rs}_{n}(G)$, $\log_{n}\operatorname{ro}_{n}(G)$, and
$\log_{n}\operatorname{rm}_{n}(G)$.

The precise value of the function $\eta(G)$ will be presented in detail in
Theorem 2.11.

We obtain upper bounds for the number of maximal subgroups of a given index in
a group in Section 2. We show that the value of the constant in the bounds of
Detomi and Lucchini [DL03] (Theorem 1.6) can be computed from Theorem B. This
will be done in Section 3. We also present a general procedure (Construction
3.3) to obtain, starting from a $d$-generated primitive group $H$ of type 1, a
$d$-generated group $G$ with all possible $G$-isomorphism classes of chief
factors isomorphic to the socle of $H$ and with associated primitive groups
isomorphic to $H$. Similar ideas (see Remark 3.5) can be used to create a
$d$-generated group with all possible crowns associated to non-Frattini
abelian chief factors of a given order whose associated primitive groups are
$d$-generated.

##  2 Upper bounds for the number of maximal subgroups of a given index in a
group

###  2.1 Bounds for the number of maximal subgroups of type 1

Let $L$ be a non-cyclic group with a unique complemented abelian minimal
normal subgroup $N$. Assume also that $N$ is complemented in $L$. As in
[DVL98], let $L^{k}$ be the direct product of $k$ copies of $L$ and set

$L_{k}=\\{(l_{1},\dots,l_{k})\in L^{k}\mid l_{1}\equiv\dots\equiv
l_{k}\pmod{N}\\}.$

Then $L_{k}$ is the product of $N^{k}$ and a diagonal subgroup

$\operatorname{diag}L^{k}=\\{(l,\dots,l)\in L^{k}\mid l\in L\\}$

of $L^{k}$. The socle of $L_{k}$ is a direct product of $k$ minimal normal
subgroups of $L_{k}$, each of them isomorphic to $N$, and that
$L_{k}/N^{k}\cong L/N$. The quotient of $L_{k+1}$ over a minimal normal
subgroup of $L_{k+1}$ is isomorphic to $L_{k}$. We have that
$\operatorname{d}(L_{1})\leq\operatorname{d}(L_{2})\leq\operatorname{d}(L_{3})\leq\dotsb$
and, according to [Luc95],
$\operatorname{d}(L_{k+1})\leq\operatorname{d}(L_{k})+1$ for every $k$. Hence,
if $d\geq\operatorname{d}(L)$, then there is a unique $k$ such that
$\operatorname{d}(L_{k})=d<\operatorname{d}(L_{k+1})$. Let us call $f(d)=k+1$.
Then, by [DVL98],

$f(d)=1+\log_{q}\bigl{(}\lvert
N\rvert^{d-1}\big{/}\lvert\operatorname{H}^{1}(L/N,N)\rvert\bigr{)},$

where $q=\lvert{\operatorname{End}_{L/N}(N)}\rvert$.

######  Theorem 2.1.

Let $U$ be a maximal subgroup of type 1 of a $d$-generated group $G$ and let
$n=\lvert G:U\rvert$. If $U$ is not normal in $G$, then the number of maximal
subgroups $M$ of $G$ such that $\operatorname{Soc}(G/M_{G})$ is $G$-isomorphic
to $\operatorname{Soc}(G/U_{G})$ is less than or equal to

$\frac{n^{d}-n\lvert\operatorname{H}^{1}(G/C,A)\rvert}{q-1},$

where $A=C/U_{G}$ is the unique minimal normal subgroup of $G/U_{G}$ and
$q=\lvert\operatorname{End}_{G/C}(A)\rvert$. If $U$ is normal in $G$, then the
number of maximal subgroups $M$ of $G$ such that $\operatorname{Soc}(G/M_{G})$
is $G$-isomorphic to $\operatorname{Soc}(G/U_{G})$ is less than or equal to

$\frac{n^{d}-1}{n-1}.$

In all cases, this number is less than or equal to $n^{d}-1$.

###### Proof.

Consider the crown $C/R$ associated to $A$, Observe that
$C/U_{G}=\operatorname{Soc}(G/U_{G})$. If

$L=[A]*G=[A]\bigl{(}G/{\operatorname{C}_{G}(C/U_{G})}\bigr{)}=[A](G/C)\cong
G/U_{G}$

is the primitive group associated to $A$ and $C/R$ is the direct product of
$k$ chief factors isomorphic to $A$, then $G/R\cong L_{k}$.

Let us suppose first that $U$ is not normal. Note that
$\operatorname{d}(L_{k})\leq d=\operatorname{d}(G)$. By the comments after
[DVL98, Theorem 2.7], we have that

$k\leq\log_{q}\frac{n^{d-1}}{\lvert\operatorname{H}^{1}(L/A,A)\rvert}.$ (1)

Let $M$ be a maximal subgroup of $G$ such that $\operatorname{Soc}(G/M_{G})$
is $G$-isomorphic to $A$. We have that $\operatorname{Soc}(G/M_{G})=C/M_{G}$
and $R\leq M_{G}\leq C$. Moreover, there exists an isomorphism $\varphi\colon
G/M_{G}\longrightarrow L$ satisfying that the composition with the projection
over $G/C$, $G/M_{G}\longrightarrow L\longrightarrow G/C$, has kernel $C$ and
gives he identity on $G/C$. Let us now give a bound for the number of possible
kernels of epimorphisms from $L_{K}$ to $L$ giving the identity on $G/C$. By
[DVL98, Lemma 2.5], the number of possible kernels of epimorphisms from
$L_{k}$ onto $L$ giving the identity on $G/C$ is $(q^{k}-1)/(q-1)$. By
Equation (1),

$\frac{q^{k}-1}{q-1}\leq\frac{\frac{n^{d-1}}{\lvert\operatorname{H}^{1}(L/A,A)\rvert}-1}{q-1}=\frac{n^{d-1}-\lvert\operatorname{H}^{1}(L/A,A)\rvert}{(q-1)\lvert\operatorname{H}^{1}(L/A,A)\rvert}.$

The number of maximal subgroups of type 1 with a given core coincides with the
number of complements of the minimal normal subgroup of the associated
primitive group, namely $n\lvert\operatorname{H}^{1}(L/A,A)\rvert$. Hence the
number of maximal subgroups $M$ of $G$ such that $\operatorname{Soc}(G/M_{G})$
is $G$-isomorphic to $A=\operatorname{Soc}(L)$ is less than or equal to

$\frac{n^{d-1}-\lvert\operatorname{H}^{1}(L/A,A)\rvert}{(q-1)\lvert\operatorname{H}^{1}(L/A,A)\rvert}\cdot
n\lvert\operatorname{H}^{1}(L/A,A)\rvert=\frac{n^{d}-n\lvert\operatorname{H}^{1}(L/A,A)\rvert}{q-1}.$

Suppose now that $A$ is central, that is, that $U$ is normal in $G=C$, then
$n$ is a prime and $G/R=C/R$ is a direct product of $k$ cyclic groups
isomorphic to $C_{n}$. Since $G$ is $d$-generated, $k\leq d$. In this case,
$q=n$ is a prime number and the number of normal subgroups $M$ of index $n$
such that $R\leq M\leq G$ is

$\frac{n^{k}-1}{n-1}\leq\frac{n^{d}-1}{n-1}.\qed$

######  Remark 2.2.

The bound of Theorem 2.1 is attained in groups like $L_{k}$, where
$L\cong\operatorname{Sym}(4)$. In this group, $q=2$, $n=4$, and
$\lvert\operatorname{H}^{1}(G/C,A)\rvert=1$. If $k=f(d)-1=2d-2$, for $d\geq
2$, we obtain that the number of maximal subgroups of index $4$ of $L_{k}$ is
exactly

$\frac{4^{d}-4}{2-1}=4^{d}-4.$

######  Corollary 2.3.

The number of maximal subgroups $M$ of type 1 and index $n=p^{r}$ of a
$d$-generated group $G$ is less than or equal to
$(n^{d}-1){\operatorname{cr}^{\mathfrak{A}}_{n}(G)}$.

###### Proof.

Given a maximal subgroup $U$ of $G$ of type 1 and index $n$, we have that its
socle $\operatorname{Soc}(G/U_{G})$ must fall into one of the $G$-isomorphism
classes of chief factors. By Theorem 2.1, we obtain that the number of maximal
subgroups $M$ of $G$ such that
$\operatorname{Soc}(G/M_{G})\cong_{G}\operatorname{Soc}(G/U_{G})$ is bounded
by $n^{d}-1$. Since the number of $G$-isomorphism classes of chief factors of
order $n$ is $\operatorname{cr}^{\mathfrak{A}}_{n}(G)$, we obtain that the
number of maximal subgroups of type 1 and index $n$ of $G$ must be bounded by
$(n^{d}-1){\operatorname{cr}^{\mathfrak{A}}_{n}(G)}$. ∎

###  2.2 Bounds for the number of maximal subgroups of type 2

The next result provides a bound for the number of maximal subgroups of type 2
of a given index of a group.

######  Theorem 2.4.

Let $G$ be a group and let $n$ be a natural number. The number of maximal
subgroups of $G$ of type 2 and index $n$ is bounded by
${\operatorname{rs}_{n}(G)}n^{2}$.

###### Proof.

Let $M$ and $U$ be two maximal subgroups of $G$ of type 2 and index $n$, By
[BBE06, Proposition 1.2.4],
$M_{G}=\operatorname{C}_{G}\bigl{(}{\operatorname{Soc}(G/M_{G})}\bigr{)}$ and
$U_{G}=\operatorname{C}_{G}\bigl{(}{\operatorname{Soc}(G/U_{G})}\bigr{)}$
coincide if, and only if, $\operatorname{Soc}(G/M_{G})$ and
$\operatorname{Soc}(G/U_{G})$ are $G$-isomorphic. Hence by the Jordan-Hölder
theorem, if $M_{G}\neq U_{G}$, then in a given chief series of $G$ there
exists a chief factor $G$-isomorphic to $\operatorname{Soc}(G/M_{G})$ and
another chief factor $G$-isomorphic to $\operatorname{Soc}(G/U_{G})$.
Consequently the number of primitive quotients of $G$ associated to a maximal
subgroup of index $n$ is $\operatorname{rs}_{n}(G)$. By Theorem 1.5, the
number of core-free maximal subgroups of index $n$ of $G/U_{G}$ for a maximal
subgroup $U$ of type 2 of $G$ is bounded by $n^{2}$, that is, the number of
maximal subgroups of index $n$ of $G$ with core $U_{G}$ is bounded by $n^{2}$.
Hence the number of maximal subgroups of $G$ of type 2 and index $n$ is
bounded by ${\operatorname{rs}_{n}(G)}n^{2}$. ∎

###  2.3 Bounds for the number of maximal subgroups of type 3

Now we analyse the maximal subgroups of type 3. The following two results show
that the minimal normal subgroups of primitive groups of type 3 fall in at
most two isomorphism classes. Although this is not needed for the proofs of
our results, we include this information here for the sake of completeness.
The first result appears as a consequence of the classification of simple
groups.

######  Lemma 2.5.

The number of non-abelian simple groups of a given order is at most $2$.

A generalisation of this result to characteristically simple groups was
formulated as a question by Cameron [Cam81], who attributed its proof to
Teague in Note (ii) at the end of his paper.

######  Theorem 2.6 (see [KLST90, Theorem 6.1]).

Let $S$ and $T$ be non-isomorphic finite simple groups. If $\lvert
S^{a}\rvert=\lvert T^{b}\rvert$ for some natural numbers $a$ and $b$, then
$a=b$ and $S$ and $T$ are either $\operatorname{PSL}_{3}(4)$ or
$\operatorname{PSL}_{4}(2)$, or are $\operatorname{O}_{2n+1}(q)$ and
$\operatorname{PSp}_{2n}(q)$ for some $n\geq 3$ and some odd $q$.

As in [JSL98], we use the non-abelian cohomology with the terminology of Serre
in [Ser64]. Given a $G$-group $B$ and a $1$-cocycle
$\beta\in\operatorname{Z}^{1}(G,B)$, then $b^{\eta(g)}=b^{gg^{\beta}}$ defines
a homomorphism $\eta\colon G\longrightarrow\operatorname{Aut}B$. The
corresponding $G$-group is denoted by $B_{\beta}$ and called the $G$-group
obtained from $B$ by torsion via $\beta$.

######  Lemma 2.7.

Let $G$ be a $d$-generated group. Let $n$ be a natural number which is a power
of the order of a non-abelian simple group. Then $\operatorname{rm}_{n}(G)\leq
n^{d}$.

###### Proof.

Consider a crown with a chief factor $A$ of order $n$ and let $B$ be another
chief factor in the same crown as $A$. We have that $A$ and $B$ are
$G$-connected (see [BBE06, Definition 1.2.5]), but not $G$-isomorphic.
According to [JSL98, Proposition 1.2], there exists a $1$-cocycle
$\beta\in\operatorname{Z}^{1}(G,B)$ such that $A$ is isomorphic to $B_{\beta}$
as $G$-groups, that is, there exists an isomorphism $\varphi\colon
A\longrightarrow B$ such that $a^{g\varphi}=a^{\varphi gg^{\beta}}$ for each
$a\in A$ and $g\in G$. Hence the number of chief factors $G$-connected to $B$
but not $G$-isomorphic to $B$ is bounded by
$\lvert\operatorname{Z}^{1}(G,B)\rvert$ and so
$\operatorname{rm}_{n}(G)\leq\lvert\operatorname{Z}^{1}(G,B)\rvert$. Since
these $1$-cocycles are uniquely determined by the image of a generating system
of $G$, we have that the number of cocycles is at most $n^{d}$. It follows
that $\operatorname{rm}_{n}(G)\leq n^{d}$. ∎

######  Theorem 2.8.

Let $G$ be a $d$-generated group and let $n$ be a natural number that is a
power of the order of a non-abelian simple group. Assume that $G$ has $s_{n}$
crowns of (non-abelian) chief factors of order $n$ that are the products of
$t_{1}$, $t_{2},\dots$, $t_{s_{n}}$ chief factors, respectively. The number of
maximal subgroups of $G$ of type 3 and index $n$ is bounded by

$n^{2}\sum_{j=1}^{s_{n}}\frac{t_{j}(t_{j}-1)}{2}.$

###### Proof.

Given a maximal subgroup $U$ of $G$ of type 3 and index $n$, the quotient
$G/U_{G}$ is a primitive group of type $3$.

The socle of a primitive quotient of type 3 of $G$ is a direct product of two
minimal normal subgroups that are $G$-isomorphic to two $G$-connected chief
factors of $G$ belonging to the same $G$-crown. Since the core of this maximal
subgroup is the intersection of the centralisers of the minimal normal
subgroups of the quotient, that are the centralisers of the chief factors
appearing in the crown, we have that the number of choices for the core is
$t_{i}(t_{i}-1)/2$, where $t_{i}$ is the number of chief factors in the
corresponding crown. This shows that the number of cores of such maximal
subgroups is bounded by $\sum_{j=1}^{s_{n}}t_{j}(t_{j}-1)/2$. By Theorem 1.5,
each of the quotients by these cores has at most $n^{2}$ core-free maximal
subgroups of index $n$. The result follows. ∎

######  Corollary 2.9.

Let $G$ be a $d$-generated group and let $n$ be a natural number which is a
power of the order of a non-abelian simple group. The number of maximal
subgroups of $G$ of type 3 and index $n$ is bounded by

$n^{2}\left(\frac{\operatorname{rm}_{n}(G)(\operatorname{ro}_{n}(G)-s_{n})}{2}\right)\leq
n^{d+2}\left(\frac{\operatorname{ro}_{n}(G)-s_{n}}{2}\right),$

where $s_{n}$ is the number of $G$-crowns of $G$ associated to non-abelian
chief factors.

###### Proof.

Since, with the notation of Theorem 2.8, we have that
$\operatorname{ro}_{n}(G)=\sum_{i=1}^{s_{n}}t_{i}$ and
$\operatorname{rm}_{n}(G)=\max_{i=1}^{s_{n}}t_{i}$, this result is an
immediate consequence of Theorem 2.8. ∎

###  2.4 General bounds

Denote by $\mathbb{T}$ the set of all prime powers greater than $1$ and by
$\mathbb{S}$ the set of all powers of the orders of non-abelian simple groups.
As a consequence of Corollary 2.3, Theorem 2.4, and Theorem 2.8, we obtain the
following conclusion.

######  Theorem 2.10.

The number ${\operatorname{m}_{n}(G)}$ of maximal subgroups of index $n$ of a
$d$-generated group $G$ satisfies the following bounds:

  1. 1.

If $n\in\mathbb{T}$, then

$\displaystyle{\operatorname{m}_{n}(G)}$
$\displaystyle\leq(n^{d}-1){{\operatorname{cr}^{\mathfrak{A}}_{n}(G)}+n^{2}{\operatorname{rs}_{n}(G)}}.$

  2. 2.

If $n\in\mathbb{S}$, then

$\displaystyle{\operatorname{m}_{n}(G)}$ $\displaystyle\leq
n^{2}{{\operatorname{rs}_{n}(G)}}+n^{2}\left(\frac{\operatorname{rm}_{n}(G)\operatorname{ro}_{n}(G)}{2}\right)$
$\displaystyle\leq
n^{2}{\operatorname{rs}_{n}(G)}+n^{d+2}\left(\frac{\operatorname{ro}_{n}(G)}{2}\right).$

  3. 3.

If $n\notin\mathbb{S}\cup\mathbb{T}$, then

${\operatorname{m}_{n}(G)}\leq n^{2}{{\operatorname{rs}_{n}(G)}}.$

We can use the bound $A+B\leq 2\max\\{A,B\\}$ to obtain the following bounds
for $\operatorname{m}_{n}(G)$:

  1. 1.

If $n\in\mathbb{T}$, then

$\displaystyle{\operatorname{m}_{n}(G)}$ $\displaystyle\leq
2\max\\{n^{d}{\operatorname{cr}^{\mathfrak{A}}_{n}(G)},n^{2}{\operatorname{rs}_{n}(G)}\\}.$

  2. 2.

If $n\in\mathbb{S}$, then

${\operatorname{m}_{n}(G)}\leq
n^{2}\max\left\\{2\operatorname{rs}_{n}(G),\operatorname{rm}_{n}(G)\operatorname{ro}_{n}(G)\right\\}.$

  3. 3.

If $n\notin\mathbb{S}\cup\mathbb{T}$, then

${\operatorname{m}_{n}(G)}\leq n^{2}{{\operatorname{rs}_{n}(G)}}.$

Therefore we can obtain the following bound for $\mathcal{M}(G)$.

$\displaystyle\mathcal{M}(G)$
$\displaystyle=\max_{n}\log_{n}\operatorname{m}_{n}(G)$
$\displaystyle\leq\max_{n}\Bigl{\\{}\max\\{d+\log_{n}2+\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G),2+\log_{n}2+\log_{n}\operatorname{rs}_{n}(G)\mid
n\in\mathbb{T}\\},$ $\displaystyle\qquad\qquad
2+\max\bigl{\\{}\log_{n}2+\log_{n}\operatorname{rs}_{n}(G),$
$\displaystyle\qquad\qquad\qquad\qquad\\!\log_{n}\operatorname{rm}_{n}(G)+\log_{n}\operatorname{ro}_{n}(G)\mid
n\in\mathbb{S}\bigr{\\}},$ $\displaystyle\qquad\qquad
2+\max\\{\log_{n}\operatorname{rs}_{n}(G)\mid
n\notin\mathbb{S}\cup\mathbb{T}\\}\Bigr{\\}}$
$\displaystyle\leq\max\Bigl{\\{}d+\max_{n\in\mathbb{T}}\\{\log_{n}2+\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G)\\},$
$\displaystyle\qquad\qquad
2+\max_{n}\\{\log_{n}2+\log_{n}\operatorname{rs}_{n}(G)\\},$
$\displaystyle\qquad\qquad
2+\max_{n\in\mathbb{S}}\bigl{\\{}\log_{n}\operatorname{rm}_{n}(G)+\log_{n}\operatorname{ro}_{n}(G)\\}\Bigr{\\}}$
$\displaystyle\leq\max\Bigl{\\{}d+\max_{n\in\mathbb{T}}\\{\log_{n}2+\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G)\\},$
$\displaystyle\qquad\qquad
2+\max_{n}\\{\log_{n}2+\log_{n}\operatorname{rs}_{n}(G)\\},$
$\displaystyle\qquad\qquad
2+d+\max_{n\in\mathbb{S}}\bigl{\\{}\log_{n}\operatorname{ro}_{n}(G)\\}\Bigr{\\}}.$

We are in a position to prove Theorem A.

###### Proof of Theorem A.

Note that $\operatorname{cr}^{\mathfrak{A}}_{n}(G)\leq r$,
$\operatorname{rs}_{n}(G)\leq r$, $\operatorname{ro}_{n}(G)\leq r$, and
$\operatorname{rm}_{n}(G)\leq n^{d}$. By Theorem 2.10, if $n\in\mathbb{T}$,
then

$\operatorname{m}_{n}(G)\leq n^{d}r+n^{2}r\leq 2n^{d}r\leq n^{d+2}r;$

if $n\in\mathbb{S}$, then

$\operatorname{m}_{n}(G)\leq n^{2}r+n^{2}rn^{d}/2\leq n^{d+2}r,$

and if $n\notin\mathbb{S}\cup\mathbb{T}$, then

$\operatorname{m}_{n}(G)\leq n^{2}r\leq n^{d+2}r.\qed$

We can obtain from Theorem 2.10 the value of the function of the upper bound
of Theorem B.

######  Theorem 2.11.

Let $G$ be a $d$-generated non-trivial group. Then, for

$\displaystyle\eta(G)$
$\displaystyle:=\max\Bigl{\\{}d+2.02+\max_{n\in\mathbb{T}}\\{\log_{n}2+\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G)\\},$
$\displaystyle\qquad\qquad\\!4.02+\max_{n}\\{\log_{n}2+\log_{n}\operatorname{rs}_{n}(G)\\},$
$\displaystyle\qquad\qquad\\!4.02+\max_{n\in\mathbb{S}}\bigl{\\{}\log_{n}\operatorname{rm}_{n}(G)+\log_{n}\operatorname{ro}_{n}(G)\\}\Bigr{\\}},$

and

$\displaystyle\kappa(G)$
$\displaystyle:=\max\Bigl{\\{}d+2.02+\max_{n\in\mathbb{T}}\\{\log_{n}2+\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G)\\},$
$\displaystyle\qquad\qquad\\!4.02+\max_{n}\\{\log_{n}2+\log_{n}\operatorname{rs}_{n}(G)\\},$
$\displaystyle\qquad\qquad\\!4.02+d+\max_{n\in\mathbb{S}}\bigl{\\{}\log_{n}\operatorname{ro}_{n}(G)\\}\Bigr{\\}},$

we have that

$\mathcal{V}(G)\leq\eta(G)\leq\kappa(G).$

######  Remark 2.12.

The terms $\log_{n}2$ in Theorem 2.11 depend on the bound $A+B\leq
2\max\\{A,B\\}$. If $A=0$ or $B=0$, then $A+B=\max\\{A,B\\}$. In particular,
in the computation of the maxima the terms $\log_{n}2$ appear only if
$n\in\mathbb{T}$ and
${\operatorname{cr}^{\mathfrak{A}}_{n}(G)}{\operatorname{rs}_{n}(G)}\neq 0$,
or $n\in\mathbb{S}$ and
${\operatorname{rs}_{n}(G)}{\operatorname{ro}_{n}(G)}\neq 0$. In all other
cases, $\log_{n}2$ can be safely removed to estimate $\mathcal{V}(G)$.

##  3 Discussion

The aim of this section is to show that the value of the constant in the upper
bound of Theorem 1.6 appears as a consequence of Theorem B and to present a
construction of groups with a large number of crowns of chief factors whose
associated primitive quotients are isomorphic to a given primitive group of
type 1.

###  3.1 Determination of the constant of Theorem 1.6

We can use the upper bound of Theorem B to determine the value of the constant
$c$ of Theorem 1.6.

######  Theorem 3.1.

The bounds of Theorem 1.6 hold with $c=5.02$.

###### Proof.

Suppose first that the number $\lambda(G)$ of non-Frattini chief factors in a
given chief series of $G$ is $1$. In this case, $G/\Phi(G)$ is the only non-
Frattini chief factor of $G$ and $G/\Phi(G)$ must be simple, either abelian or
non-abelian. If $G/\Phi(G)$ is abelian, then $G$ is cyclic and $G$ has exactly
one crown of non-Frattini abelian chief factors. Therefore $\mathcal{V}(G)\leq
d+\log_{2}2+2.02=d+3.02$ (note that the term $\log_{2}2$ can even be safely
removed from this sum). Assume now that $G/\Phi(G)$ is non-abelian. It follows
that $\operatorname{rs}_{n}(G)=1$, where $n$ is the index of a maximal
subgroup of $G/\Phi(G)$ and so $\mathcal{V}(G)\leq 4.02+\log_{5}2<5.02$ (we
could even remove the term $\log_{5}2$). In both cases,
$\mathcal{V}(G)\leq\lfloor d+3.02\rfloor$.

Now assume that $\lambda(G)>1$. It will be enough to show that for a constant
$c$, all three elements whose maximum is being considered are bounded by
$d+c\log\lambda(G)$ for a constant $c$. Note also that
$\operatorname{cr}^{\mathfrak{A}}_{n}(G)$, $\operatorname{rs}_{n}(G)$,
$\operatorname{ro}_{n}(G)$, and $\operatorname{rs}_{n}(G)$ are less than or
equal to $\lambda(G)$. The first term satisfies

$\displaystyle
d+2.02+\max_{n}\\{\log_{n}2+\log_{n}\operatorname{cr}^{\mathfrak{A}}_{n}(G)\\}$
$\displaystyle\leq d+2.02+1+\log\lambda(G)$ $\displaystyle\leq
d+4.02\log\lambda(G).$

The second term satisfies the inequality

$\displaystyle 4.02+\max_{n}\\{\log_{n}2+\log_{n}\operatorname{rs}_{n}(G)\\}$
$\displaystyle\leq 4.02+\log_{5}2+\frac{\log\lambda(G)}{\log 5}$
$\displaystyle\leq d+\frac{2.02+\log_{5}2+1}{\log 5}\log\lambda(G)$
$\displaystyle\leq d+2.31\log\lambda(G)$

(recall that we can assume $d\geq 2$ because $G$ is not cyclic). Finally, the
third term satisfies

$4.02+d+\max_{n\in\mathbb{S}}\bigl{\\{}\log_{n}\operatorname{ro}_{n}(G)\\}\leq
4.02+d+\lambda(G)$

Therefore, Theorem 1.6 holds with $c=5.02$. ∎

###  3.2 Some examples and constructions

We conclude this section by presenting a couple of examples in which our bound
improves substantially the previously known bounds.

In the following we will construct some groups with many crowns of chief
factors whose associated primitive quotients are isomorphic to a given
monolitic primitive group. This construction will depend on a a general
construction for subdirect products.

######  Construction 3.2.

Suppose that $G_{i}=\langle a_{i1},\dots,a_{id}\rangle$, $1\leq i\leq r$ are
$d$-generated groups. Let $D=\prod_{i=1}^{r}G_{i}$ be the direct product of
the groups $G_{i}$, $1\leq i\leq r$. Consider $a_{j}=\prod_{i=1}^{r}a_{ij}\in
D$, $1\leq j\leq d$. Let $\hat{G}=\langle a_{1},\dots,a_{r}\rangle\leq D$. We
have that the restriction to $\hat{G}$ of the natural projection
$\pi_{i}\colon D\longrightarrow G_{i}$, $1\leq i\leq r$, is a group
epimorphism with kernel $\hat{G}\cap\prod_{k\neq i}G_{k}$. Hence each chief
factor $H_{i}/K_{i}$ of $G_{i}$ is isomorphic to a chief factor $H/K$ of
$\hat{G}$. Furthermore, $G_{i}$ acts on $H_{i}/K_{i}$ as $\hat{G}$ acts on
$H/K$. Moreover, $\bigcap_{i=1}^{r}(\hat{G}\cap\prod_{k\neq i}G_{k})=1$, and
so $\hat{G}$ is a subdirect product of the subgroups $G_{i}$. Consequently,
each chief factor of $\hat{G}$ is isomorphic to a chief factor of one of the
$G_{i}$. However, it might happen that the number of chief factors of
$\hat{G}$ is less than the number of chief factors of all the $G_{i}$. For
instance, this happens if $G_{i}=\langle a_{1i}\rangle\cong C_{p}$, $1\leq
i\leq 2$: in this case, $\hat{G}\cong C_{p}$.

The following construction for $d$-generated groups with all possible crowns
whose associated primitive quotients are isomorphic to a given monolitic
primitive group is motivated by an argument of Hall [Hal36].

######  Construction 3.3.

Let $G$ be a $d$-generated group. We can consider the set $\Omega$ of all
generating ordered $d$-tuples of $G$. The group $\operatorname{Aut}G$ acts on
$\Omega$ in the natural way. This decomposes $\Omega$ into orbits for the
action of $\operatorname{Aut}G$, $\Omega=\Omega_{1}\cup\dots\cup\Omega_{r}$
say. Let $(a_{i1},\dots,a_{id})$ be an element of the orbit $\Omega_{i}$,
$1\leq i\leq r$. We can apply Construction 3.2 to $G_{i}=\langle
a_{i1},\dots,a_{id}\rangle=G$, $1\leq i\leq r$, to obtain $\hat{G}=\langle
a_{1},\dots,a_{d}\rangle$. If $G$ is a primitive group of type 1 and socle
$N$, then the socle of $\hat{G}$ is the product of all faithful and
irreducible modules for $G/N$ such that the corresponding primitive group is
isomorphic to $G$.

######  Example 3.4.

There are three isomorphism classes of $2$-generated primitive groups of type
1 with socle of order $8$, namely $G_{1}=[C_{2}^{3}]C_{7}$,
$G_{2}=[C_{2}^{3}][C_{7}]C_{3}$ and
$G_{3}=[C_{2}^{3}]{\operatorname{GL}_{3}(2)}$. We can construct $2$-generated
groups $\hat{G}_{1}$, $\hat{G}_{2}$, and $\hat{G}_{3}$ with Construction 3.3.
By using Construction 3.2, we can construct a subdirect product $S$ of
$\hat{G}_{1}$, $\hat{G}_{2}$, and $\hat{G}_{3}$ in such a way the generating
pairs of all these three groups are identified.

An application of this construction with GAP [GAP21] shows that

$\displaystyle\operatorname{cr}^{\mathfrak{A}}_{8}(\hat{G}_{1})$
$\displaystyle=\operatorname{cr}^{\mathfrak{A}}_{8}(\hat{G}_{2})=16,$
$\displaystyle\operatorname{cr}^{\mathfrak{A}}_{8}(\hat{G}_{3})$
$\displaystyle=114,$
$\displaystyle\operatorname{cr}^{\mathfrak{A}}_{7}(\hat{G}_{1})$
$\displaystyle=1,$
$\displaystyle\operatorname{cr}^{\mathfrak{A}}_{7}(\hat{G}_{2})$
$\displaystyle=8,$
$\displaystyle\operatorname{cr}^{\mathfrak{A}}_{3}(\hat{G}_{2})$
$\displaystyle=1,$
$\displaystyle\operatorname{r}_{\operatorname{GL}_{3}(2)}(\hat{G}_{3})$
$\displaystyle=57;$

the other values of the ranks and the numbers of abelian crowns are zero. We
note that if $1\leq i<j\leq 3$, two chief factors of $S$ corresponding to
chief factors of $\hat{G}_{i}$ and $\hat{G}_{j}$, respectively, cannot be
$S$-isomorphic, because the groups of automorphisms induced by $S$ on the
chief factors in the primitive groups $G_{i}$ and $G_{j}$ are not isomorphic.
Therefore $\operatorname{cr}^{\mathfrak{A}}_{3}(S)=1$,
$\operatorname{cr}^{\mathfrak{A}}_{7}(S)=9$,
$\operatorname{cr}^{\mathfrak{A}}_{8}(S)=146$,
$\operatorname{r}_{\operatorname{GL}_{3}(2)}(S)=57$; the other values are zero
(we note that, in fact, $S$ coincides with the direct product
$\hat{G}_{1}\times\hat{G}_{2}\times\hat{G}_{3}$). Since the indices of the
maximal subgroups of $\operatorname{GL}_{3}(2)$ are $7$ and $8$ (see for
instance [CCN+85]), we conclude that
$\operatorname{rs}_{7}(S)=\operatorname{rs}_{8}(S)=57$ and
$\operatorname{ro}_{168}(S)=\operatorname{rm}_{168}(S)=57$. The crowns of
chief factors of order $8$ in $\hat{G}_{1}$ are minimal normal subgroups, in
$\hat{G}_{2}$ are products of three minimal normal subgroups, while in
$\hat{G}_{3}$ are products of two minimal normal subgroups. The crowns of
chief factors of order $7$ in $\hat{G}_{1}$ and $\hat{G}_{2}$ coincide with
the corresponding chief factors. There is a unique crown composed of two
central chief factors of order $3$. Hence $S$ has $16+16\cdot 3+114\cdot
2=292$ chief factors of order $8$, $8$ chief factors of order $7$, $2$ chief
factors of order $3$, and $57$ chief factors isomorphic to
$\operatorname{GL}_{3}(2)$.

By Theorem B we obtain the bound

$\displaystyle\mathcal{V}(S)\leq\max\bigl{\\{}$ $\displaystyle
d+2.02+\log_{3}2+\log_{3}\operatorname{cr}^{\mathfrak{A}}_{3}(S),$
$\displaystyle
d+2.02+\log_{7}2+\log_{7}\operatorname{cr}^{\mathfrak{A}}_{7}(S),$
$\displaystyle
d+2.02+\log_{8}2+\log_{8}\operatorname{cr}^{\mathfrak{A}}_{8}(S),$
$\displaystyle 4.02+\log_{7}2+\log_{7}\operatorname{rs}_{7}(S),$
$\displaystyle 4.02+\log_{8}2+\log_{8}\operatorname{rs}_{8}(S),$
$\displaystyle
4.02+\log_{168}\operatorname{rm}_{168}(G)+\log_{168}\operatorname{ro}_{168}(G)\bigr{\\}},$

that is,

$\displaystyle\mathcal{V}(S)\leq\max\\{$ $\displaystyle
4.02+\log_{3}2+\log_{3}1,$ $\displaystyle 4.02+\log_{7}2+\log_{7}9,$
$\displaystyle 4.02+\log_{8}2+\log_{8}146,$ $\displaystyle
4.02+\log_{7}2+\log_{7}57,$ $\displaystyle 4.02+\log_{8}2+\log_{8}57,$
$\displaystyle 4.02+2\log_{168}57\\}\leq 6.75.$

The arguments of Lubotzky would take into account the $2$ abelian chief
factors of order $3$ to obtain the upper bound for $\operatorname{m}_{3}(S)$;
the $8$ abelian chief factors of order $7$ and the $57$ non-abelian chief
factors isomorphic to $\operatorname{GL}_{3}(2)$ to obtain the upper bound for
$\operatorname{m}_{7}(S)$; the $292$ abelian chief factors of order $8$ and
the $57$ chief factors isomorphic to $\operatorname{GL}_{3}(2)$ to obtain the
upper bound for $\operatorname{m}_{8}(S)$, and the $57$ chief factors
isomorphic to $\operatorname{GL}_{3}(2)$ and order $168$ to obtain the upper
bound for $\operatorname{m}_{168}(S)$. We obtain the following bounds:

$\displaystyle\operatorname{m}_{3}(S)$ $\displaystyle\leq((1/2)(0+1)\cdot
0+2\cdot 3^{2})3^{2}=162,$ $\displaystyle\operatorname{m}_{7}(S)$
$\displaystyle\leq((1/2)(57+1)\cdot 57+8\cdot 7^{2})7^{2}=100\,205,$
$\displaystyle\operatorname{m}_{8}(S)$ $\displaystyle\leq((1/2)(57+1)\cdot
57+292\cdot 8^{2})8^{2}=1\,301\,824,$ $\displaystyle\operatorname{m}_{168}(S)$
$\displaystyle\leq((1/2)(57+1)\cdot 57+0\cdot 168^{2})168^{2}=46\,654\,272$

and so we conclude that

$\displaystyle\mathcal{V}(S)$ $\displaystyle\leq
2.02+\max\\{\log_{3}(162),\log_{7}(100\,205),$
$\displaystyle\qquad\qquad\qquad\qquad\log_{8}(1\,301\,824),\log_{168}(46\,654\,272)$
$\displaystyle=2.02+\log_{8}(1\,301\,824)\leq 8.791.$

The bounds obtained for the numbers of maximal subgroups of each index with
the application of our Theorem 2.10 are:

$\displaystyle\operatorname{m}_{3}(S)$
$\displaystyle\leq(3^{2}-1)\operatorname{cr}^{\mathfrak{A}}_{3}(S)=8,$
$\displaystyle\operatorname{m}_{7}(S)$
$\displaystyle\leq(7^{2}-1)\operatorname{cr}^{\mathfrak{A}}_{7}(S)+7^{2}\operatorname{rs}_{7}(S)=48\cdot
9+49\cdot 57=3\,225,$ $\displaystyle\operatorname{m}_{8}(S)$
$\displaystyle\leq(8^{2}-1)\operatorname{cr}^{\mathfrak{A}}_{8}(S)+8^{2}\operatorname{rs}_{8}(S)=63\cdot
146+64\cdot 57=12\,846,$ $\displaystyle\operatorname{m}_{168}(S)$
$\displaystyle\leq
168^{2}\left(\frac{\operatorname{rm}_{168}(S)(\operatorname{ro}_{168}(S)-s_{168})}{2}\right)$
$\displaystyle=168^{2}\cdot\left(\frac{57\cdot 56}{2}\right)=45\,045\,504.$

In fact, Theorem 2.1 gives the exact value
$\operatorname{m}_{3}(S)=(3^{2}-1)/(3-1)=4$.

######  Remark 3.5.

The procedure of Example 3.4 can be followed with all isomorphism classes of
$d$-generated monolitic primitive groups with a given socle to obtain a group
with all possible crowns of non-Frattini chief factors of a given order. It is
enough to consider all isomorphism classes of $d$-generated monolitic
primitive groups whose socle has a given order, to use Construction 3.3 with
each of them, and then to use Construction 3.2 with the obtained groups.

## Acknowledgements

These results are part of the R+D+i project supported by the Grant
PGC2018-095140-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A
way of making Europe”, as well as by the Grant PROMETEO/2017/057 funded by
GVA/10.13039/501100003359, and partially supported by the Grant E22 20R,
funded by Departamento de Ciencia, Universidades y Sociedad del Conocimiento,
Gobierno de Aragón/10.13039/501100010067.

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