Intersection of maximal subgroups in prosolvable groups

TL;DR

This paper explores the structure of prosolvable groups by examining how intersections of maximal subgroups, called η-intersections, influence the group's properties and what they reveal about the group's composition.

Contribution

It introduces the concept of η-intersections in prosolvable groups and analyzes their implications for the group's structural characteristics.

Findings

Characterization of η-intersections in prosolvable groups

Relationship between η-intersections and group structure

Insights into subgroup intersection properties

Abstract

Let $H$ be an open subgroup of a profinite group that can be expressed as intersection of maximal subgroups of $G.$ Given a positive real number $\eta,$ we say that $H$ is an $\eta$-intersection if there exists a family of maximal subgroups $M_1,\ldots,M_t$ such that $H=M_1\cap\ldots\cap M_t$ and $|G:M_1|\cdots|G:M_t|\le |G:H|^{\eta}$. We investigate the meaning of this property and its influence on the group structure.

Equations

$$P(G,s)=\sum _H\mu (H,G)|G:H{|}^{-s},$$

$$\underset{n\to \infty }{\mathrm{liminf}}\frac{{\tilde{\beta }}_n(G)}{{\tilde{\gamma }}_n(G)}=0.$$

$$1+2+\dots +n^{\eta }=\frac{n^{\eta }(n^{\eta }+1)}{2}$$

$$c_n(G)\le \frac{n^{\eta }(n^{\eta }+1)}{2}{n}^{\eta \cdot \alpha }.$$

$$\frac{n^{\eta }(n^{\eta }+1)}{2}{n}^{\eta \cdot \alpha }\le n^{\beta },$$

$$R_G(A)=\bigcap _{M\in {\mathcal{M}}_V}M.$$

$$X^{w_1}\le (H^{v_1}{)}^{w_1}=H^{v_1+w_1}=H^{v_2+w_2},$$

$$(W_1\cap W_2)X^{w_1}$$

$$K\cap M=W_{1}X\cap W_2{H}^{v_2}=W_{1}X\cap W_{2}X\cap W_2{H}^{v_2}.$$

$$K\cap M=W_{1}X\cap W_{2}X\cap W_2{H}^{v_2}=W_{1}X\cap W_{2}X=W_{1}X=K$$

$$M=W_2{H}^{v_2}=W_2{H}^{v_2-w}=W_2(H^{v_1}{)}^u.$$

$$\bigcap _{i=1}^n{M}_i=U{C}_{H^{\ast }}(Z)$$

$$V^t>U_1>U_2>\dots >U_{t^{\ast }}=\dots =U_n=U$$

$$M_1\cap \cdots \cap M_j=U_j{H}^{w_j}.$$

$$M_1\cap \cdots \cap M_j=K\cap M_j=U_j{H}^{v_j}$$

$$M_1\cap \cdots \cap M_{t^{\ast }}=U{H}^{\ast }$$

$$M_1\cap \cdots \cap M_{t^{\ast }}\cap \cdots \cap M_{t^{\ast }+i}=U{C}_{H^{\ast }}(Z_i).$$

$$M_1\cap \cdots \cap M_{t^{\ast }+i}=K\cap M_{t^{\ast }+i}=M{C}_{C_{H^{\ast }}(Z_{i-1})}(z_i)=M{C}_{H^{\ast }}(Z_i),$$

$$(X_1\cap \cdots \cap X_{t^{\ast }})\cap (Y_1\cap \cdots \cap Y_d)=(U{H}^{\ast })\cap (A{C}_{H^{\ast }}(Z))=U{C}_{H^{\ast }}(Z).\text{qed}$$

$$H=X_1\cap \dots \cap X_{\rho }\cap Y_1\cap \dots \cap Y_{\sigma },$$

$$X:=X_1\cap \dots \cap X_{\rho }\text{ and }Y:=Y_1\cap \dots \cap Y_{\sigma }.$$

$$G/R=C/R⋊K/R\cong V^t⋊K/R,$$

$$X/R=T/R⋊C_{K^{\ast }}(Z)/R.$$

$$C_{K^{\ast }}(Z)=C_{K^{\ast }}(Z^{\ast })\cap ({\cap }_{M\in {\mathcal{M}}_Z}M)$$

$${\tilde{X}}_1/R\cap \cdots \cap {\tilde{X}}_{\alpha }/R=T/R⋊C_{K^{\ast }}(Z^{\ast })/R.\text{ i.e. }{\tilde{X}}_1\cap \cdots \cap {\tilde{X}}_{\alpha }=T{C}_{K^{\ast }}(Z^{\ast }).$$

$$\left({\cap }_{1\le i\le \alpha }{\tilde{X}}_i\right)\cap \left({\cap }_{1\le j\le \beta }{\tilde{Y}}_j\right)=\left(T{C}_{K^{\ast }}(Z^{\ast })\right)\cap \left(C\left({\cap }_{1\le j\le \beta }{M}_j\right)\right)=T{C}_{K^{\ast }}(Z)=X,$$

$${\tilde{X}}_1\cap \cdots \cap {\tilde{X}}_{\alpha }\cap {\tilde{Y}}_1\cap \cdots \cap {\tilde{Y}}_{\beta }=X.$$

$$\tilde{X}={\tilde{X}}_1\cap \cdots \cap {\tilde{X}}_{\alpha },\tilde{Y}={\tilde{Y}}_1\cap \cdots \cap {\tilde{Y}}_{\beta }\cap Y_1\cap \cdots \cap Y_{\sigma }.$$

$$\tilde{Y}=Q_1\cap \cdots \cap Q_{\tau }\text{ and }\prod _{1\le j\le \tau }|G:Q_j|\le |G:\tilde{Y}{|}^{\gamma +1}.$$

$$V^{t-t^{\ast }}{\cong }_{G}T/R{\cong }_G{D}^{\ast }R/R{\cong }_G{D}^{\ast }/(D^{\ast }\cap R){\cong }_G{D}^{\ast }.$$

Full text

1. Introduction

Let $G$ be a (topologically) finitely generated profinite group. We write $c_{n}(G)$ to express the number of (open) subgroups of $G$ of index $n$ which are intersections of maximal subgroups of $G.$ We will say that $c_{n}(G)$ is polynomially bounded if there exists $\beta$ independent of $n$ such that $c_{n}(G)\leq n^{\beta}$. Mann asked the following question [5, Problem 4]:

Question 1**.**

What are the groups for which $c_{n}(G)$ is polynomially bounded?

This question is related with the discussion of a conjecture proposed by Mann in the same paper. A profinite group $G$ is positively finitely generated (PFG) if for some finite $k,$ a random $k$-tuple of elements generates $G$ with probability $P(G,k)>0.$ Mann conjectured that if $G$ is PFG, then the Dirichlet series

[TABLE]

defines an analytic function on some right half-plane of $\mathbb{C}$ and takes the values $P(G,k)$ for (sufficiently large) $k\in\mathbb{N}$ (here $H$ ranges over the lattice of all open subgroups of $G$ and $\mu$ is the Möbius function associated to this lattice). To establish this, it is sufficient to verify that $P(G,s)$ converges absolutely in some right half-plane; this is the case if and only if

1. (1)

$|\mu(H,G)|$ is bounded by a polynomial function of $|G:H|;$ 2. (2)

the number $b_{n}(G)$ of open subgroups $H$ of index $n$ satisfying $\mu(H,G)\neq 0$ grows at most polynomially in $n.$

As it is noticed in [5] p. 447, if $H$ is an open subgroup of $G$ and $\mu(H,G)\neq 0$, then $H$ is an intersection of maximal subgroups, hence $b_{n}(G)\leq c_{n}(G)$ and condition (2) is satisfied if $c_{n}(G)$ is polynomially bounded. A celebrated result of Mann and Shalev [7] states that a profinite group is PFG if and only if it has polynomial maximal subgroup growth (PMSG). Since $\mu(M,G)=-1$ for any maximal subgroup $M$ of $G,$ it must be $m_{n}(G)\leq b_{n}(G)\leq c_{n}(G)$ (where $m_{n}(G)$ denotes the number of maximal subgroups of $G$ with index $n$). In particular, if $c_{n}(G)$ grows polynomially, then $G$ has PMSG. A natural question that arises from these considerations is the following:

Question 2**.**

Is $c_{n}(G)$ polynomially bounded when $G$ has PMSG?

In [4], Mann’s conjecture about the absolutely convergence of $P(G,s)$ has been proved in the particular case when $G$ is a finitely generated prosolvable groups (recall that every finitely generated prosolvable group $G$ has PMSG [5, Theorem 10]). In that paper it is proved that if $G$ is a finitely generated prosolvable, then $b_{n}(G)$ is polynomially bounded but is still unknown whether $c_{n}(G)$ is also polynomially bounded. In this paper, we want to investigate whether the methods employed in [4] to prove that $b_{n}(G)$ is polynomially bounded, can be adapted to study the behaviour of $c_{n}(G).$ The main observation in [4] is that if $G$ is a prosolvable group and $H$ is an open subgroup of $G$ with $\mu(H,G)\neq 0,$ then the maximal subgroups $M_{1},\ldots,M_{t}$ such that $H=M_{1}\cap\ldots\cap M_{t}$ can be chosen with the extra property that $|G:M_{1}|\cdots|G:M_{t}|=|G:H|.$ On the other hand, what is really useful to bound the sequence $b_{n}(G)$ is not the equality $|G:M_{1}|\cdots|G:M_{t}|=|G:H|$ but that existence of a constant $\eta$, independent on the choice of $H,$ such that $|G:M_{1}|\cdots|G:M_{t}|\leq|G:H|^{\eta}.$ This suggests the following definitions:

Definition 1**.**

Let $G$ be a finite group and let $\eta$ be a positive real number. We say that a maximal intersection $H$ in $G$ is an $\eta$-intersection if there exists a family of maximal subgroups $M_{1},\ldots,M_{t}$ such that:

1. (1)

$H=M_{1}\cap\ldots\cap M_{t}$. 2. (2)

$|G:M_{1}|\cdots|G:M_{t}|\leq|G:H|^{\eta}$.

Definition 2**.**

Let $G$ be a finite group and let $\eta$ be a positive real number. We say that $G$ has the $\eta$-intersection property if every maximal intersection in $G$ is an $\eta$-intersection.

Definition 3**.**

Let $G$ be a profinite group. We say that $G$ has the bounded intersection property if there exists a positive real number $\eta$ such that every open subgroup $H$ of $G$ which is an intersection of maximal subgroups of $G$ is an $\eta$-intersection.

Clearly a profinite group $G$ has the bounded intersection property if and only if there exists a positive real number $\eta$ such that $G/N$ has the $\eta$-intersection property for every open normal subgroup $N$ of $G.$

The connection between the previous definitions and Question 1 is clarified by the following Proposition.

Proposition 4**.**

Suppose that $G$ is a profinite group with polynomial maximal subgroup growth. If $G$ has the bounded intersection properties, then there exists a constant $\beta$ such that $c_{n}(G)\leq n^{\beta}.$

Other two definitions will play a relevant role in our investigation of the bounded intersection property.

Definition 5**.**

Let $G$ be a finite group, $V$ an irreducible $G$-module, $F=\operatorname{End}_{G}(V)$ and $\gamma$ a positive integer. We say that $V$ is a $\gamma$-module if, for every $F$-subspace $W$ of $V,$ there exists an $F$-subspace $W^{*}$ of $V$ such that:

1. (1)

$\dim_{F}(W^{*})\leq\gamma.$** 2. (2)

$C_{G}(W)=C_{G}(W^{*})\cap(\cap_{M\in\mathcal{M}_{W}}M)$* where $\mathcal{M}_{W}$ is the set of the maximal subgroups of $G$ containing $C_{G}(W).$*

Definition 6**.**

Let $G$ be a profinite group. We say that $G$ has the bounded chief factors property if the exists a positive integer $\gamma$ such that every complemented chief factor of $G$ is a $\gamma$-module.

Let $M$ be a maximal subgroup of $G$ and denote by $Y_{M}=\bigcap_{g\in G}M^{g}$ the normal core of $M$ in $G$ and by $X_{M}/Y_{M}$ the socle of the primitive permutation group $G/Y_{M}$ (in its action on the right cosets of $M/Y_{M}$ in $G/Y_{M}$): clearly $X_{M}/Y_{M}$ is a chief factor of $G$ and $M/Y_{M}$ is a complement of $X_{M}/Y_{M}$ in $G/Y_{M}.$ Now assume that $H$ is a maximal intersection in $G$: we will denote by $\mathcal{V}_{H}$ the set of the irreducible $G$-module which are isomorphic to $X_{M}/Y_{M}$ for some maximal subgroup $M$ of $G$ containing $H.$ We will prove:

Theorem 7**.**

Let $G$ be a finite solvable group and let $H$ be a maximal intersection in $G$. If every $V\in\mathcal{V}_{H}$ is a $\gamma$-module, then $H$ has the $(\gamma+1)$-intersection property.

Corollary 8**.**

Let $G$ be a finite solvable group. If there exists $\gamma$ such that every complemented chief factor of $G$ is a $\gamma$-module, then $G$ has the $(\gamma+1)$-intersection property.

We will also prove a converse result:

Theorem 9**.**

Let $G$ be a finite solvable group. If $G$ has the $\eta$-intersection property, then every complemented chief factor is a $\lfloor\eta\cdot c\rfloor$-module, with $c\backsimeq 3.243,$ the Pálfy-Wolf constant.

In particular we deduce:

Theorem 10**.**

A prosolvable group $G$ has the bounded intersection property if and only if it has the bounded chief factors property.

Clearly if $V$ is an irreducible $G$-module and $\dim_{\operatorname{End}_{G}(V)}V\leq\gamma,$ then $V$ is a $\gamma$-module. So the following corollary is an immediate consequence of Proposition 4 and Theorem 10.

Corollary 11**.**

Let $G$ be a finitely generated prosolvable. If there exists $\gamma\in\mathbb{N}$ such that $\dim_{\operatorname{End}_{G}(V)}V\leq\gamma$ for every irreducible $G$-module $G$-isomorphic to a complemented chief factor of $G,$ then $c_{n}(G)$ is polynomially bounded.

An application of the previous result, is the following:

Corollary 12**.**

Let $G$ be a finitely generated prosolvable group. If the derived subgroup is pronilpotent, then $c_{n}(G)$ is polynomially bounded.

In particular, we may apply Corollary 12 to the class of the finitely generated prosupersolvable groups. However, even if the number of intersections of maximal subgroups in a finitely generated prosupersolvable group $G$ grows polynomially with respect to the index, it may happen that the amount of such subgroups is really “big” comparing with the number of subgroups of $G$ with non-zero Möbius number. Let us denote by $\tilde{\beta}_{n}(G)$ the number of conjugacy classes of subgroups with index at most $n$ and with non-zero Möbius number and by $\tilde{\gamma}_{n}(G)$ the number of conjugacy classes of subgroups with index at most $n$ that are intersection of maximal subgroups. In Section 4 we will construct a 2-generated prosupersolvable group with the properly that

[TABLE]

We don’t know examples of prosolvable groups that don’t satisfy the bounded chief factors property. A positive answer to the following intriguing question, will imply that all the prosolvable groups have the bounded chief factors property.

Question 3**.**

Does there exists a constant $\gamma$ such that, for every finite solvable group $G$, all the irreducible $G$-modules are $\gamma$-modules?

Notice that we can strengthen the definition of $\gamma$-module, setting that an irreducible $G$-module $V$ is a strong $\gamma$-module if, for every $\operatorname{End}_{G}(V)$-subspace $W$ of $V,$ there exists an $\operatorname{End}_{G}(V)$-subspace $W^{*}$ of $V$ such $\dim_{\operatorname{End}_{G}(V)}(W^{*})\leq\gamma$ and $C_{G}(W)=C_{G}(W^{*})$.

Question 4**.**

Does there exists a constant $\gamma$ such that, for every finite solvable group $G$, all the irreducible $G$-modules are strongly $\gamma$-modules?

2. Preliminary results

In this section we will give the proof of Proposition 4 and we will recall the main properties of the crowns of a finite solvable group. Moreover we will start to study the maximal intersections in a relevant case.

Proof of Proposition 4.

Since $G$ has PMSG, there exists $\alpha$ such that, for each $n\in\mathbb{N}$, the number of maximal subgroups of $G$ with index $n$ is bounded by $n^{\alpha}.$ Now, for $n\neq 1$, we want to count the number of subgroups $H$ with $|G:H|=n$ which are intersections of maximal subgroups. By assumption there exists an integer $\eta$ with the property that, if we fix such an $H\leq G$, then there is a family of maximal subgroups $M_{1},\ldots,M_{t}$ such that $H=\cap_{1\leq i\leq t}M_{i}$ and $n_{1}\ldots n_{t}\leq n^{\eta}$, where $n_{i}=|G:M_{i}|$. There are at most

[TABLE]

possible factorizations of positive integers $\leq n^{\eta}$ (see [6]), and for each fixed factorization $n_{1}\ldots n_{t}$, there are at most $n_{i}^{\alpha}$ choices for the maximal subgroup $M_{i}$ of index $n_{i}$. Therefore, there are at most $n_{1}^{\alpha}\ldots n_{t}^{\alpha}\leq n^{\eta\cdot\alpha}$ choices for the family $M_{1},\ldots,M_{t}$, and we conclude that

[TABLE]

Obviously, we always can find a constant $\beta$ such that

[TABLE]

for any $n\geq 1$, so the proof is complete. ∎

Let $G$ be a finite solvable group and let $M$ be a maximal subgroup of $G$ and denote by $Y_{M}=\bigcap_{g\in G}M^{g}$ the normal core of $M$ in $G$ and by $X_{M}/Y_{M}$ the socle of the primitive permutation group $G/Y_{M}$: clearly $X_{M}/Y_{M}$ is a chief factor of $G$ and $M/Y_{M}$ is a complement of $X_{M}/Y_{M}$ in $G/Y_{M}.$ Let $\mathcal{M}$ be the set of maximal subgroups of $G,$ let $\mathcal{V}$ be a set of representatives of the irreducible $G$-modules that are $G$-isomorphic to some chief factor of $G$ having a complement and, for every $V\in\mathcal{V},$ let $\mathcal{M}_{V}$ be the set of maximal subgroups $M$ of $G$ with $X_{M}/Y_{M}\cong_{G}V.$

Recall some results by Gaschütz [3]. Given $V\in\mathcal{V}$, let

[TABLE]

It turns out that $R_{G}(A)$ is the smallest normal subgroup contained in $C_{G}(A)$ with the property that $C_{G}(A)/R_{G}(A)$ is $G$-isomorphic to a direct product of copies of $A$ and it has a complement in $G/R_{G}(A)$. The factor group $C_{G}(A)/R_{G}(A)$ is called the $A$-crown of $G$. The non-negative integer $\delta_{G}(A)$ defined by $C_{G}(A)/R_{G}(A)\cong_{G}A^{\delta_{G}(A)}$ is called the $A$-rank of $G$ and it coincides with the number of complemented factors in any chief series of $G$ that are $G$-isomorphic to $A$ (see for example [1, Section 1.3]). In particular $G/R_{G}(A)\cong A^{\delta_{G}(A)}\rtimes H,$ with $H\cong G/C_{G}(A).$

Lemma 13**.**

[1, Lemma 1.3.6]** Let $G$ be a finite solvable group with trivial Frattini subgroup. There exists a crown $C/R$ and a non trivial normal subgroup $D$ of $G$ such that $C=R\times D.$

Lemma 14**.**

[2, Proposition 11]** Assume that $G$ is a finite solvable group with trivial Frattini subgroup and let $C,R,D$ be as in the statement of Lemma 13. If $HD=HR=G,$ then $H=G.$

For a fixed $V\in\mathcal{V},$ we want to study which subgroups of $G$ can be obtained as intersection of maximal subgroups in $\mathcal{M}_{V}.$ Since $R_{G}(V)\leq M$ for every $M\in\mathcal{M}_{V},$ we are indeed asking which subgroups of the semidirect product $G/R_{G}(A)\cong V^{\delta_{G}(H)}\rtimes G/C_{G}(V),$ can be obtained as intersection of maximal supplements of the socle $V^{\delta_{G}(H)}.$ So assume that $H$ is a solvable group acting irreducibly and faithfully on an elementary abelian $p$-group $V$ and, for a positive integer $t,$ consider the semidirect product $G=V^{t}\rtimes H$, where we assume that the action of $H$ is diagonal on $V^{t},$ that is, $H$ acts in the same way on each of the direct factors. The aim of the remaining part of this section is to describe which subgroups of $G$ can be obtained as intersections of maximal subgroups of $G$ supplementing $V^{t}.$ Notice that $M$ is a maximal in $G$ supplementing $V^{t}$ if and only if $M=WH^{v}$ with $W$ a maximal $H$-submodule of $V^{t}$ and $v\in V^{t}$.

Lemma 15**.**

Let $G=V^{t}\rtimes H$ be a finite solvable group such that $V$ is a faithful irreducible $H$-module. Let $K=W_{1}X$ and $M=W_{2}H^{v_{2}}$, where $W_{1}$ is a proper $H$-submodule of $V^{t}$, $W_{2}$ is maximal submodule of $V^{t},$ $X\leq H^{v_{1}}$ for some $v_{1}\in V^{t}$ and $v_{2}\in V^{t}.$

1. i)

If $W_{1}+W_{2}=V^{t},$ then there exists $w_{1}\!\in\!W_{1}$ such that $K\cap M=(W_{1}\cap W_{2})X^{w_{1}}.$ 2. ii)

If $W_{1}\leq W_{2},$ then there exists $u\in V$ such that $K\cap M=W_{1}C_{X}(u).$

Proof.

Assume $W_{1}+W_{2}=V^{t}$. There exist $w_{1}\in W_{1}$ and $w_{2}\in W_{2}$ such that $v_{1}-v_{2}=w_{2}-w_{1}$, or equivalently, $v_{1}+w_{1}=v_{2}+w_{2}$. Thus, we have

[TABLE]

In particular

[TABLE]

Moreover, let $y_{1}x=y_{2}h$ be an element of $K\cap M$ with $y_{1}\in W_{1}$, $x\in X^{w_{1}}$, $y_{2}\in W_{2}$ and $h\in H^{v_{2}+w_{2}}$. We have $y_{1}-y_{2}=hx^{-1}\in V^{t}\cap H^{v_{2}+w_{2}}=1$, so it follows that $y_{1}=y_{2}$ and $h=x$. Hence, $K\cap M\leq(W_{1}\cap W_{2})X^{w_{1}}$, and since we proved above the other inclusion, we have the equality, as we wanted.

Assume now that $W_{1}\leq W_{2}$. We observe that

[TABLE]

If $W_{2}X\leq W_{2}H^{v_{2}}$, then

[TABLE]

and we are done. So, we may assume $W_{2}X\not\leq W_{2}H^{v_{2}}$. This, in particular, implies $W_{2}H^{v_{1}}\neq W_{2}H^{v_{2}}$ and consequently, $v_{2}-v_{1}\not\in W_{2}$. Since $V^{t}$ is a completely reducible $H$-module, there exists an $H$-submodule $U$ of $V^{t}$ such that $V^{t}=W_{2}\times U$. Thus, there exists a non-trivial element $u\in U$ such that $v_{2}-v_{1}=w+u$ with $w\in W_{2}$. Hence, $H^{v_{2}-w}=H^{v_{1}+u}$ and

[TABLE]

Let us show that $K\cap M=W_{1}C_{X}(u)$. Note that $C_{X}(u)=C_{X}(u)^{u}\leq(H^{v_{1}})^{u}$, so it is clear that $W_{1}C_{X}(u)\leq K\cap M$. In order to prove the other inclusion let $w_{1}x=w_{2}h^{u}$ be an element of $K\cap M$ with $w_{1}\in W_{1}$, $w_{2}\in W_{2}$, $x\in X$ and $h\in H^{v_{1}}$. Note that $w_{1}x=w_{2}h^{u}=w_{2}[u,h^{-1}]h$, and since $H$ normalizes $U$, it follows that $[U,H]\leq U$. So, $w_{1}-w_{2}-[u,h^{-1}]=hx^{-1}\in V^{t}\cap H^{v_{1}}=1$, or in other words, $w_{1}-w_{2}=[u,h^{-1}]$ and $h=x$. Moreover, $w_{1}-w_{2}=[u,h^{-1}]\in W_{2}\cap U=0$, so $w_{1}=w_{2}$ and $[u,h^{-1}]=[u,h]=[u,x]=0$. Therefore, $x\in C_{X}(u)$ and $K\cap M\leq W_{1}C_{X}(u)$, so that $K\cap M=W_{1}C_{X}(u)$, and we are done. ∎

Theorem 16**.**

Let $G=V^{t}\rtimes H$ be a finite solvable group such that $V$ is a faithful irreducible $H$-module. Let $M_{1}=W_{1}H^{v_{1}},\ldots,M_{n}=W_{n}H^{v_{n}}$ be maximal subgroups of $G$ supplementing $V^{t}$, with $W_{i}$ a maximal $H$-submodule of $V^{t}$ and $v_{i}\in V^{t}$ for every $i$. Then, we have

[TABLE]

where $U=\bigcap_{i=1}^{n}W_{i}$, $H^{*}$ is a conjugate of $H$ and $Z$ is an $\operatorname{End}_{H}(V)$-subspace of $V$.

Proof.

Set $U_{j}:=\bigcap_{1\leq i\leq j}W_{i}$. Reordering the maximal subgroups, we may assume

[TABLE]

for a suitable $1\leq t^{*}\leq n$. First we prove, by induction on $j$, that for every $j\leq t^{*}$ there exists $w_{j}\in V^{t}$ such that

[TABLE]

This is clear if $j=1.$ Assume $1<j\leq t^{*}.$ By induction $K:=M_{1}\cap\dots\cap M_{j-1}=U_{j-1}H^{w_{j-1}}$ for some $w_{j-1}\in V^{t}.$ Since $U_{j}=U_{j-1}\cap W_{j}<U_{j-1},$ we have $U_{j-1}+W_{j}=V^{t}$ and therefore, by Lemma 15,

[TABLE]

for some $w_{j}\in V^{t}.$ In particular

[TABLE]

with $H^{*}$ a suitable conjugate of $H$ and $U\cong_{H}V^{t-t^{*}}.$ Now consider $0\leq i\leq n-t^{*}.$ We prove, again by induction, that, for every $0\leq i\leq n-t^{*},$ there exists an $\operatorname{End}_{H}(V)$-subspace $Z_{i}$ of $V$ with

[TABLE]

When $i=0,$ we just take $Z_{0}=\{0\}.$ If $i>0,$ then by induction there exists $Z_{i-1}$ such that $K=M_{1}\cap\dots\cap M_{t^{*}}\cap\dots\cap M_{t^{*}+i-1}=UC_{H^{*}}(Z_{i-1}).$ Since $U\leq W_{t^{*}+i},$ it follows from Lemma 15 that there exists $z_{i}\in V$ such that

[TABLE]

being $Z_{i}$ the $\operatorname{End}_{H}(V)$-subspace of $V$ spanned by $Z_{i-1}$ and $z_{i}.$ ∎

Theorem 17**.**

Let $G=V^{t}\rtimes H$ be a finite solvable group such that $V$ is a faithful irreducible $H$-module. Assume that $U$ is an intersection of maximal $H$-submodules of $V^{t}$, $H^{*}$ is a conjugate of $H$ and $Z$ is an $\operatorname{End}_{H}(V)$-subspace of $V.$ If $V^{t}/U\cong_{H}V^{t^{*}}$ and $d=\dim_{\operatorname{End}_{H}(V)}Z,$ then $UC_{H^{*}}(Z)$ can be obtained as intersection of $t^{*}+d$ maximal subgroups of $G$ supplementing $V^{t}.$

Proof.

By assumption, there exists $t^{*}$ maximal $H$-submodules $W_{1},\dots,W_{t^{*}}$ of $V^{t}$ such that $U=W_{1}\cap\dots\cap W_{t^{*}}.$ Let $z_{1},\dots,z_{d}$ be an $\operatorname{End}_{H}(V)$-basis of $Z.$ Let $A$ be a maximal $H$-submodule of $V^{t}$ containing $U:$ there exists an $H$-submodule $B$ of $V^{t}$ such that $V^{t}=A\times B.$ It must be $B\cong_{H}V,$ so let $\phi:V\to B$ be an $H$-isomorphism and for every $1\leq i\leq d,$ set $b_{i}:=z_{i}^{\phi}.$ For every $1\leq i\leq t^{*}$, let $X_{i}:=W_{i}H^{*}$ and, for every $1\leq j\leq d$, let $Y_{j}:=A(H^{*})^{b_{j}}.$ It follows from Lemma 15, that

[TABLE]

3. Bounded intersection and bounded chief factors properties

In this section we will prove that in the class of prosolvable groups the bounded intersection property and the bounded chief factors property are equivalent.

Proof of Theorem 7.

We proceed by induction on $|G|$. We may assume $\operatorname{Frat}(G)=1$. By Lemma 13, we can assume there exists an irreducible $G$-module $V$ such that $C=R\times D$ with $1\neq D\cong_{G}V^{t}$, where $C=C_{G}(V)$, $R=R_{G}(V)$ and $t=\delta_{G}(V)$. Assume that $H$ is a maximal intersection in $G$ and that every $U\in\mathcal{V}_{H}$ is a $\gamma$-module. We want to prove that there exists a family $M_{1},\dots,M_{n}$ of maximal subgroups of $G$ such that $H=M_{1}\cap\cdots\cap M_{n}$ and $|G:M_{1}|\cdots|G:M_{n}|\leq|G:H|^{\gamma+1}.$

First assume $D\leq H.$ If $M\in\mathcal{M}_{V},$ then $MD=G,$ hence $H\not\leq M$ and consequently $V\notin\mathcal{V}_{H}.$ We may then work module $D$ and conclude by induction. Hence we may assume $D\not\leq H$. We can write

[TABLE]

where $X_{1},\dots,X_{\rho}$ are maximal subgroups not containing $D$ and $Y_{1},\dots,Y_{\sigma}$ are maximal subgroups containing $D.$ Notice also that $\{X_{1},\dots,X_{\rho}\}\subseteq\mathcal{M}_{V},$ and consequently $V\in\mathcal{V}_{H}.$ We define

[TABLE]

By Lemma 14, $R\leq X_{i}$ for every $i$, in particular $R\leq X$. By the properties of the crowns, there exists $K\leq G$ such that

[TABLE]

where $V$ can be seen as an irreducible $K$-module with $C_{K}(V)=R.$ Note that $X/R$ is an intersection of maximal subgroups of $G/R$ supplementing $C/R\cong_{G}V^{t}$, so we may apply Theorem 16: there exists an intersection $T/R$ of maximal $G$-submodules of $C/R,$ a conjugate $K^{*}$ of $K$ in $G$ and an $\operatorname{End}_{G}(V)$-subspace $Z$ of $V$ such that

[TABLE]

Since $V\in\mathcal{V}_{H},$ there is an $\operatorname{End}_{G}(V)$-subspace $Z^{*}$ of $Z$ such that $\dim_{\operatorname{End}_{G}(V)}Z^{*}\leq\gamma$ and

[TABLE]

where $\mathcal{M}_{Z}$ is the set of the maximal subgroups of $K^{*}$ containing $C_{K^{*}}(Z).$ We have $C/T\cong_{G}V^{t^{*}}$ for some positive integer $t^{*}$ and, by Theorem 17, there exists $\alpha\leq t^{*}+\gamma$ and a family $\tilde{X}_{1},\dots,\tilde{X}_{\alpha}$ of maximal subgroups of $G$ containing $R$ and supplementing $C$ such that

[TABLE]

Assume now that $\mathcal{M}_{Z}=\{M_{1},\dots,M_{\beta}\}:$ for every $1\leq i\leq\beta,$ there exists a maximal subgroup $\tilde{Y}_{i}$ containing $C$ such that $\tilde{Y}_{i}/R=C/R\rtimes M_{i}/R.$ Notice that

[TABLE]

hence

[TABLE]

Let

[TABLE]

We have $\tilde{X}\cap\tilde{Y}=X\cap Y=H.$ Notice that $\tilde{Y}$ is an intersection of maximal subgroups of $G$ containing $D$, so by induction there exists $\tau$ maximal subgroups $Q_{1},\dots,Q_{\tau}$ of $G$ such that

[TABLE]

Define $D^{*}=D\cap X$. Thus, by the Dedekind Law, we have $T=DR\cap\tilde{X}=(D\cap\tilde{X})R=D^{*}R$, and note that

[TABLE]

Moreover $\tilde{Y}\geq D$ implies $\tilde{X}\tilde{Y}\geq\tilde{X}D$, and so $|\tilde{X}\tilde{Y}|\geq|\tilde{X}D|$. Hence,

[TABLE]

so

[TABLE]

We have

[TABLE]

and, since $|G:X_{i}|=|V|$ for every $i\in\{1,\dots,\alpha\}$, $\alpha\leq t^{*}+\gamma$ and $t^{*}\geq 1,$

[TABLE]

and the theorem follows. ∎

Proof of Theorem 9.

Assume that $V$ is an irreducible $G$-module $G$-isomorphic to a non-Frattini chief factor of $G$ and let $H:=G/C_{G}(V).$ The semidirect product $\Gamma=V\rtimes H$ is an epimorphic image of $G$, so it has the $\eta$-intersection property. Let $F=\operatorname{End}_{G}(V)$ and let $W$ be an $F$-subspace of $V.$ By Theorem 17, $X=C_{H}(W)$ is an intersection of maximal subgroups of $\Gamma,$ so there exists $n$ maximal subgroups $X_{1},\dots,X_{n}$ of $\Gamma$ such that $X=X_{1}\cap\dots\cap X_{n}$ and $|\Gamma:X_{1}|\cdots|\Gamma:X_{n}|\leq|\Gamma:X|^{\eta}.$ Let $\overline{X}$ be the intersection of the maximal subgroups of $H$ containing $X.$ We may assume that $X_{i}$ contains $V$ if and only if $i>m:$ this means that for every $j\leq m$ there exists $v_{j}\in V$ such that $X_{j}=H^{v_{j}}$ while $X_{m+1}\cap\dots\cap X_{n}\geq V\rtimes\overline{X}.$ In particular

[TABLE]

which implies that if $W^{*}$ is the $F$-subspace of $V$ spanned by $v_{1},\dots,v_{m}$, then $C_{H}(W)=C_{H}(W^{*})\cap(\cap_{M\in\mathcal{M}_{W}}M).$ By [8] and [9], $|\Gamma|\leq|V|^{c},$ so

[TABLE]

Hence $\dim_{F}(W^{*})\leq m\leq\lfloor\eta\cdot c\rfloor.$ ∎

It remains to prove Corollary 12. For this purpose, we need the following observation:

Lemma 18**.**

Assume that $G$ is a finite solvable group with nilpotent derived subgroup. If $V$ is an irreducible $G$-module $G$-isomorphic to a non-Frattini chief factor of $G,$ then $\dim_{\operatorname{End}_{G}V}V=1.$

Proof.

We may assume $\operatorname{Frat}(G)=1.$ This means that $G=M\rtimes H$ where $H$ is abelian and $M=V_{1}\times\cdots\times V_{u}$ is the direct product of $u$ irreducible non trivial $H$-modules $V_{1},\dots,V_{u}.$ Let $F_{i}=\operatorname{End}_{H}(V_{i})=\operatorname{End}_{G}(V_{i})$: for each $i\in\{1,\dots,u\},$ $V_{i}$ is an absolutely irreducible $F_{i}H$-module so $\dim_{F_{i}}V_{i}=1.$ Now assume that $A$ is a nontrivial irreducible $G$-module $G$-isomorphic to a complemented chief factor of $G$: it must be $A\cong_{G}V_{i}$ for some $i$, so $|\operatorname{End}_{G}(A)|=|F_{i}|=|V_{i}|=|A|.$ ∎

Proof of Corollary 12.

Let $G$ be a finitely generated prosolvable group with pronilpotent derived subgroup. By Lemma 18, every non-Frattini chief factor of $G$ is a 1-module, so, by Theorem 7, $G$ has the 2-intersection property. The conclusion follows from Proposition 4. ∎

4. Prosupersolvable groups

Recall that the Dirichlet Theorem on arithmetic progressions states that for any two positive coprime integers $a$ and $b$, there exist infinitely many primes which are congruent to $a$ modulo $b$. In particular, the arithmetic progression $\{1+r\cdot 2^{n}\mid r\in\mathbb{N}\}$ contains infinitely many primes. This implies that there exists a strictly ascending sequence $\{p_{n}\}_{n\in\mathbb{N}}$ of primes with the property that $2^{n}$ divides $p_{n}-1$ and $p_{n+1}>2^{n}\cdot p_{1}\cdots p_{n}.$

Let $V_{m}$ be a 1-dimensional vector space over $\operatorname{F}_{m}$, where $\operatorname{F}_{m}$ is the field with $p_{m}$ elements, and $H_{n}:=\langle x_{n}\rangle$ be a cyclic group of order $2^{n}$ . We can define an action of $H_{n}$ on $V_{m}$, for every $m\leq n$, as follows: if $v\in V_{m}$, then $v^{x_{n}}:=\zeta_{m}v$, where $\zeta_{m}$ is an element of order $2^{m}$ in $\operatorname{F}_{m}^{*}$. Note that $C_{H_{n}}(V_{m})=\langle x_{n}^{2^{m}}\rangle$. Consider the following finite supersolvable group:

[TABLE]

Let us describe the maximal subgroups of $G_{n}$. Let $W=V_{1}\times\ldots\times V_{n}$ and for every $1\leq i\leq n,$ set $W_{i}:=V_{1}\times\ldots\times V_{i-1}\times V_{i+1}\times\ldots\times V_{n}.$ First note that $M=W\rtimes\langle x_{n}^{2}\rangle$ is the unique maximal subgroup of $G$ containing $W$. The other maximal subgroups of $G_{n}$ are the semidirect products $W_{i}\rtimes H_{n}^{v}$ for $i\in\{1,\dots,n\}$ and $v_{i}\in V$, so we have precisely $p_{i}$ maximal subgroups of index $p_{i}$ for every $1\leq i\leq n$. Consider now the maximal subgroups $M_{1}=W_{i}\rtimes H_{n}^{v_{1}}$ and $M_{2}:=W_{i}\rtimes H_{n}^{v_{2}}$ with $v_{1},v_{2}\in V_{i}$. If $v_{1}\neq v_{2}$, then $M_{1}\neq M_{2}$ and, by Lemma 15, $M_{1}\cap M_{2}=W_{i}\rtimes\langle x_{n}^{2^{i}}\rangle,$ since $\langle x_{n}^{2^{i}}\rangle=C_{H_{n}}(v)$ for every $0\neq v\in V_{i}.$ These considerations imply that a subgroup $H$ of $G_{n}$ can be expressed as intersection of maximal subgroups of $G$ if and only if it is conjugated to one of the following subgroups:

1. (1)

$X_{J}=\left(\prod_{j\notin J}V_{j}\right)\rtimes H_{n},$ where $J$ is a non-empty subset of $\{1,\dots,n\};$ note that $|G_{n}:X_{J}|=\prod_{j\in J}p_{j}.$ 2. (2)

$Y_{J}=\left(\prod_{j\notin J}V_{j}\right)\rtimes\langle x_{n}^{2}\rangle,$ where $J$ is a non-empty subset of $\{1,\dots,n\};$ note that $|G_{n}:Y_{J}|=2\cdot\prod_{j\in J}p_{j}.$ 3. (3)

$Z_{J,i}=\left(\prod_{j\notin J}V_{j})\right)\rtimes\langle x_{n}^{2^{i}}\rangle,$ where $i\in\{2,\dots,n\}$ and $J$ is a subset of $\{1,\dots,n\}$ containing $i$. Note that $|G_{n}:Y_{Z,i}|=2^{i}\cdot\prod_{j\in J}p_{j}.$

By [4, Theorem 1.5], if $K$ is a subgroup of $G$ with $\mu(K,G_{n})\neq 0$, then there exists a family of maximal subgroups $M_{1},\ldots,M_{t}$ of $G_{n}$ such that $K=M_{1}\cap\cdots\cap M_{t}$ and $|G_{n}:K|=|G_{n}:M_{1}|\cdots|G_{n}:M_{t}|$. The subgroups $Z_{J,i}$ do not have this property, hence $\mu(Z_{J,i},G_{n})=0$ for all the possible choices of $J$ and $i.$ It follows that, for $n\geq 2$,

[TABLE]

and consequently

[TABLE]

Now consider the inverse limit $G=\varprojlim_{n}G_{n}.$ Note that $G$ is a 2-generated prosupersolvable group, with $G\cong(\prod_{i}V_{i})\rtimes\mathbb{Z}_{2},$ being $\mathbb{Z}_{2}$ the group of the 2-adic integer. Let $n^{*}=2^{n}\cdot p_{1}\cdots p_{n}.$ The condition $p_{n+1}>2^{n}\cdot p_{1}\cdots p_{n}$ for every $n\in\mathbb{N},$ implies that if $H$ is an open subgroup of $G$ with $|G:H|>n^{*},$ then $\left(\prod_{j>n}V_{j}\right)\rtimes\mathbb{Z}_{2}^{2^{n}}\leq H$ and consequently $\tilde{\beta}_{n^{*}}(G)=\tilde{\beta}_{n^{*}}(G_{n})$ and $\tilde{\gamma}_{n^{*}}(G)=\tilde{\gamma}_{n^{*}}(G_{n}).$ This implies

[TABLE]