1 Introduction
Throughout this paper, all groups are finite. Let G be a noncyclic group. A cover of G is a family
$\{X_i\}_{i=1}^{n}$
of proper subgroups such that
$G = \bigcup _{i=1}^{n} \, X_i$
. A cover is said to be irredundant if none of the subgroups can be omitted. A cover is called a partition of G if
$X_i \cap X_j = \{1\}$
whenever
$i \neq j$
. It is natural to ask what information about G can be deduced from properties of the subgroups
$X_i$
and their number n.
Interest in covers of groups by proper subgroups began with the classical work of Scorza [Reference Scorza12], which showed that a group G admits an irredundant cover by exactly three subgroups when
$G/N \cong C_{2}\times C_{2}$
for some normal subgroup N. Later, Cohn [Reference Cohn4] introduced the invariant
$\sigma (G)$
, defined as the minimal number of subgroups in an irredundant cover of a noncyclic group, and classified the groups G with
$\sigma (G)=3,4,5$
. He also noted that no example with
$\sigma (G)=7$
seemed to exist and conjectured that every noncyclic solvable group satisfies
$\sigma (G)=p^{a}+1$
. Both statements were confirmed by Tomkinson [Reference Tomkinson15].
A cyclic subgroup H of G is called a maximal cyclic subgroup if it is not properly contained in any other cyclic subgroup of G. If G is noncyclic, then its maximal cyclic subgroups form an irredundant cover. In [Reference Rogério11], Rogério introduced another cover invariant defined as follows:
$\lambda (G)$
denotes the maximum size of an irredundant cover of a group G. Rogério observed that the number of maximal cyclic subgroups of G coincides with
$\lambda (G)$
. Groups G with
$\lambda (G) \leqslant 6$
are classified in [Reference Bastos, Lima and Rogério1, Reference Rogério11]. In addition, [Reference Bastos, Lima and Rogério1] shows that if
$\lambda (G) \leqslant 30$
, then G is solvable. See also [Reference Brodie3, Reference Garonzi and Lucchini7] for related work.
This work is motivated by several recent contributions. Qureshi and Reis [Reference Qureshi and Reis9] introduced flower groups, while Gao and Garonzi [Reference Gao and Garonzi6] and Zarrin [Reference Zarrin17] obtained structural bounds in terms of certain special covers. We refine these ideas, incorporating results of Cohn [Reference Cohn4] and Rogério [Reference Rogério11], and obtain new bounds for the derived length and order of a group in terms of its maximal cyclic subgroups. Regarding the derived length, we follow the method used in [Reference Gao and Garonzi6] and improve the estimate by at least one.
Theorem 1.1. Let G be a noncyclic solvable group with
$\lambda (G)=\lambda $
. Then, the derived length is at most
$$ \begin{align*} \begin{cases} 2 & \text{if } \ \lambda \leqslant 6, \\ \tfrac{5}{2} \cdot \log_3(\lambda-1)+1 & \text{if } \lambda \geqslant 7.\\ \end{cases} \end{align*} $$
Note that the bound is exact for
$\lambda (G)\leqslant 6$
;
$\mathsf {SL}(2, 3)$
has derived length 3 and
$\lambda (\mathsf {SL}(2, 3))=7$
. In Table 1, we use GAP [13] to list some solvable groups, illustrating how far the bound appears to be from the precise value. We also establish a similar bound for another type of cover (see Theorem 3.4).
Table 1 Examples of groups with derived length at most 6.
Let G be a finite noncyclic group with
$\lambda (G) = \lambda $
and let
$S=\{\mathcal {C}_1,\ldots ,\mathcal {C}_{\lambda }\}$
be the set of its maximal cyclic subgroups. The group G is called a flower group if there exists a subgroup
$\mathcal {C}_0\leqslant G$
such that
$$ \begin{align*} \mathcal{C}_i\cap \mathcal{C}_j=\mathcal{C}_0 \quad (1\leqslant i<j\leqslant \lambda). \end{align*} $$
Here,
$\mathcal {C}_0$
is the pistil and the subgroups in S are the petals [Reference Qureshi and Reis9]. Note that if G is a flower group with trivial pistil, then the family
$\{\mathcal {C}_1,\ldots ,\mathcal {C}_{\lambda }\}$
is a partition of G. For a deeper discussion of partitionable groups, we refer the reader to [Reference Zappa16].
A group G is said to be an
$\mathsf {M}_{\mathcal {C}}$
-group if every proper maximal cyclic subgroup of G is a maximal subgroup and G has at least one maximal cyclic subgroup (see [Reference Juriaans and Rogério8]). In the next result, we establish a connection between
$\mathsf {M}_{\mathcal {C}}$
-groups and flower groups.
Theorem 1.2. Let G be a noncyclic
$\mathsf {M}_{\mathcal {C}}$
-group. Then, G is a flower group.
Our next result gives a bound for
$|G|$
in terms of
$\lambda (G)$
and
$\exp (G)$
. These parameters are independent (see Remark 3.12(1)).
Theorem 1.3. Let G be a noncyclic group with
$\lambda (G)=\lambda $
.
-
(1) The order
$|G|$
is at most
$\exp (G) \cdot (\lambda -1)$
. -
(2) Let G be a flower group with pistil
$\mathcal {C}_0$
. Then, each petal
$\mathcal {C}_i$
has order at most
$|\mathcal {C}_0| \cdot (\lambda -1)$
. Moreover,
$|G| \leqslant |\mathcal {C}_0| \cdot (\lambda -1)^2$
. -
(3) Let G be a flower p-group with pistil
$\mathcal {C}_0$
. Then,
$\exp (G) \leqslant |\mathcal {C}_0| \cdot (\lambda -1)$
.
The bounds obtained above are sharp. For instance, the quaternion group
$G = Q_8$
is a flower group with
$\exp (G)=4$
,
$\lambda (G)=3$
and
$|\mathcal {C}_0|=2$
. See Remarks 3.11 and 3.12 for further details on Theorem 1.3.
We say that a family of groups is maximal cyclic bounded
$\mathsf {MCB}$
if for every natural number n, there are only finitely many groups G in the family with
$\lambda (G) \leqslant n$
(see [Reference Gao and Garonzi6, Section 1] for more details). Flower groups are not
$\mathsf {MCB}$
(see Example 3.9). However, by restricting the size of the pistil, we obtain
$\mathsf {MCB}$
families. For each positive integer k, let
$\mathfrak {X}_k$
denote the set of all flower groups G with
$|\mathcal {C}_0| \leqslant k$
. The following result is immediate from Theorem 1.3(2).
Corollary 1.4. For every k, the family
$\mathfrak {X}_k$
is
$\mathsf {MCB}$
.
2 Preliminaries
Throughout, we will use the usual notation, for example,
$\mathsf {S}_{n}$
and
$\mathsf {A}_{n}$
denote respectively the symmetric and alternating groups on n letters, and
$D_n$
stands for the dihedral group of order
$2n$
. Let H be a subgroup of a group G. The subgroup
$\bigcap _{g \in G} H^g$
is called the core of H in G and is denoted by
$H_G$
. The derived length of a solvable group G will be denoted by
$\mathsf {dl}{(}G)$
.
We begin by recalling a few known results that will be used throughout the paper.
Proposition 2.1 (Rogério [Reference Rogério11, Propositions 4 and 5]).
Let G be a group.
-
(a) If
$H_1, \ldots , H_n$
are the maximal cyclic subgroups of G, then
$\{H_{i}\}_{i=1}^{n}$
is an irredundant cover of G and
$\lambda (G)=n$
. Moreover,
$\{H_{i}\}_{i=1}^{n}$
is the unique covering by proper subgroups of G if and only if
$H_i$
is a maximal subgroup of G for
$i=1,\ldots , n$
. -
(b) If
$M \trianglelefteq G$
, then
$\lambda (G/M) \leqslant \lambda (G)$
. If
$N=\bigcap _{i=1}^{\lambda }H_{i}$
, where
$ \lambda =\lambda (G)$
and
$H_{1},\,H_{2},\ldots ,\,H_{\lambda }$
are the maximal cyclic subgroups of G, then N is a central subgroup of G and
$\lambda (G/N)=\lambda (G)$
.
Lemma 2.2 (Cohn [Reference Cohn4, Theorem 1]).
Let G be a group. If
$\{X_{i}\}_{i=1}^{n}$
is an irredundant cover of G, then
$\vert G\vert \leqslant \sum _{i=2}^{n}\vert X_{i}\vert .$
The following result is a direct consequence of [Reference Tomkinson14, Lemma 3.3].
Lemma 2.3. Let G be a noncyclic group with
$\lambda (G)=\lambda $
and let
$H_1,\ldots ,H_\lambda $
be its maximal cyclic subgroups ordered so that
$|H_i|\geqslant |H_{i+1}|$
. Then,
$|G:H_j|\le \lambda -1$
for
$j=1,2$
.
3 Proofs
3.1 Proof of Theorem 1.1
In [Reference Bastos, Lima and Rogério1, Theorems A and B] and [Reference Rogério11, Theorem 1], the authors give explicit presentations for all groups with
$\lambda (G) \in \{3,4,5,6\}$
. By direct inspection of these presentations, we obtain the following result.
Lemma 3.1. Let G be a nonabelian group with
$\lambda = \lambda (G) \leqslant 6$
. Then, G is metabelian.
Proof of Theorem 1.1.
By Lemma 3.1, it suffices to consider
$\lambda \geqslant 7$
. Let
$H_1,\ldots ,H_{\lambda }$
be the maximal cyclic subgroups of G, ordered so that
$|H_i|\geqslant |H_{i+1}|$
for each i. By Proposition 2.1(a), their union forms an irredundant cover of G. By Lemma 2.3, the index of
$H_1$
in G is at most
$\lambda -1$
and the quotient
$G/(H_1)_G$
embeds into the symmetric group
$\mathsf {S}_{\lambda -1}$
. Dixon’s theorem [Reference Dixon5, Theorem 1] then shows that the derived length of this quotient is at most
$(5/2)\log _3(\lambda -1)$
. As the core of
$H_1$
is cyclic, the stated estimate follows immediately.
Remark 3.2. Here,
$[m,n]$
denotes the SmallGroup(m,n) in the GAP library [13]. In Table 1, we have used GAP to select a solvable group for each derived length at most
$6$
, computed
$\lambda (G)$
and listed the bound obtained from Theorem 1.1 for comparison (see also [Reference Gao and Garonzi6, Theorem 1.1]).
Remark 3.3. Our methods do not yield good bounds for the derived length
$\mathsf {dl}{(}G)$
in terms of
$\lambda (G)$
. Roughly speaking, the approach fails because it relies on Tomkinson’s result [Reference Tomkinson14] which gives very large bounds whose precise values are unknown. A different strategy is needed.
3.2 Derived length and centralisers
The main result of this subsection lies slightly outside the main scope of the paper. However, its application is natural and arises from the same ideas employed in the proof of Theorem 1.1.
Let G be a group. Here,
$\mathsf {cent}(G)$
denotes the set
$\{C_G(g) \, | \, g \in G\},$
where
$C_G(g)$
is the centraliser of the element g in G. For a positive integer, the group G is called an n-centraliser group if
$|\mathsf {cent}(G)| = n$
. Zarrin [Reference Zarrin17] proves that if G is a solvable n-centraliser group, then
$\mathsf {dl}{(}G) \leqslant n$
. Using the same ideas, together with Dixon’s theorem [Reference Dixon5, Theorem 1], we obtain the following bound.
Theorem 3.4. Let G be a solvable n-centraliser group. Then,
$$ \begin{align*} \mathsf{dl}{(}G) \leqslant \begin{cases} 2 & \text{if } n \leqslant 5,\\ \tfrac{5}{2} \cdot \log_3(n-1)+2 & \text{if } n \geqslant 6.\\ \end{cases} \end{align*} $$
Before giving the proof, we present two lemmas about n-centraliser groups.
Lemma 3.5 (Belcastro and Sherman [Reference Belcastro and Sherman2, Theorems 2 and 4, page 367]).
-
(a) There are no
$2$
-centraliser and no
$3$
-centraliser groups. -
(b) A group G is a
$4$
-centraliser group if and only if
$ G/Z(G) = C_2 \times C_2$
. -
(c) A group G is a
$5$
-centraliser group if and only if
$ G/Z(G) = C_3 \times C_3$
or
$S_3$
.
Lemma 3.6 (Zarrin [Reference Zarrin17, Lemma 2.2]).
Let G be an n-centraliser group. Let
$$ \begin{align*} N := \bigcap_{a\in G} N_G(C_G(a)). \end{align*} $$
Then,
$G/N$
is isomorphic to a subgroup of
$\mathsf {S}_{n-1}$
.
Proof of Theorem 3.4.
Assume that
$n \leqslant 5$
. By Lemma 3.5, the group
$G/Z(G)$
is abelian or metacyclic. Note that if
$G/Z(G)$
is metacyclic, then G is metabelian, since a central-by-cyclic group is abelian.
Now, suppose
$n \geqslant 6$
. By Lemma 3.6, the quotient
$G/N$
embeds in
$\mathsf {S}_{n-1}$
. By Dixon’s result [Reference Dixon5, Theorem 1], the derived length of
$G/N$
is at most
$(5/2)\log _3(n-1)$
. Hence, it suffices only to prove that N is metabelian.
First, we prove that N is
$2$
-Engel. Choose arbitrarily
$a,b \in N$
. We claim that
$[b,a,a]=1$
. In fact, by definition,
$$ \begin{align*} (a^{-1})^b \in C_G(a)^b = C_G(a). \end{align*} $$
Hence,
$(a^{-1})^b a = [b,a] \in C_G(a)$
and so
$[b,a,a]=1$
. This shows that N is
$2$
-Engel. According to Levi’s theorem [Reference Robinson10, Theorem 12.3.6](iv), we deduce that N is nilpotent of class at most
$3$
. In particular, N is metabelian, which completes the proof.
Remark 3.7. There is no general relation between
$\lambda (G)$
and
$|\mathsf {cent}(G)|$
. Indeed, for the dihedral group
$D_n$
, we have
$\lambda (D_n) = n+1$
, while
$$ \begin{align*} |\mathsf{cent}(D_n)| = \begin{cases} n+2 & \text{if } n \text{ is odd}, \\ \tfrac{1}{2}{n} + 2 & \text{if } n \text{ is even}. \end{cases} \end{align*} $$
3.3 Proof of Theorem 1.2
First, we state the classification of finite
$\mathsf {M}_{\mathcal {C}}$
-groups obtained by Juriaans and Rogério [Reference Juriaans and Rogério8].
Lemma 3.8 (Juriaans and Rogério [Reference Juriaans and Rogério8, Theorem 2.5]).
Let G be a finite group. Then, G is an
$\mathsf {M}_{\mathcal {C}}$
-group if and only if one of the following holds:
-
(a) G is cyclic or
$G \cong C_{p} \times C_{p} \times C_{n}$
, where p is a prime not dividing n; -
(b)
$G \cong C_{n} \times Q_{8}$
with n odd, where
$Q_{8} = \langle x,y \mid x^4 = 1, x^2 = y^2, x^y = x^{-1}\rangle $
is the quaternion group of order
$8$
; -
(c) there exist primes
$q < p$
, an exact sequence and
$$ \begin{align*} 1 \to Z(G) \to G \to C_{p} \rtimes C_{q} \to 1, \end{align*} $$
$Z(G) \leqslant \langle x\rangle $
for any maximal cyclic group
$\langle x\rangle \leqslant G$
.
Proof of Theorem 1.2.
The result follows by inspecting the three cases in Lemma 3.8.
Case (a):
$G = C_p \times C_p \times C_n \cong C_p \times C_{pn}$
with
$p \nmid n$
. By Proposition 2.1(b),
$\lambda (G)=p+1$
. We can write G as
$\langle a,b \mid a^{p}=b^{pn}=1,\ [a,b]=1\rangle $
. The maximal cyclic subgroups are
$H_i = \langle a^{i}b\rangle $
for
$i \in \{1,\ldots ,p\}$
and
$H_{p+1} = \langle ab^{p}\rangle $
. Moreover,
$H_i \cap H_j = \langle b^{p}\rangle $
for all
$ i \neq j$
. Since
$\langle b^{p}\rangle $
has order n and is the common intersection of all distinct maximal cyclic subgroup, G is a flower group, which gives item (a).
Case
$(b)$
:
$G = Q_8 \times C_n$
, where n is odd. We can rewrite G as
$$ \begin{align*}\langle a,x,y \ \mid \ a^n = x^4 = [a,x] = [a,y]=1, x^2 = y^2, x^y=x^{-1}\rangle.\end{align*} $$
By [Reference Rogério11],
$\lambda (G)=3$
. The maximal cyclic subgroups of G are
$H_1 = \langle ay \rangle $
,
$H_2 = \langle ax \rangle $
and
$H_3 = \langle axy \rangle $
, and
$H_i \cap H_j = \langle x^2a\rangle $
whenever
$i \neq j$
. Hence, G is a flower group with pistil
$\langle x^2a\rangle $
, which gives item (b).
Case (c): there exist primes p and q such that G fits into the short exact sequence
$$ \begin{align*} 1 \to Z(G) \to G \to C_{p} \rtimes C_{q} \to 1, \end{align*} $$
and every maximal cyclic subgroup
$\langle x\rangle $
of G contains the centre. Let
$H_{1},\ldots ,H_{\lambda }$
be the maximal cyclic subgroups of G. By Proposition 2.1(b),
$$ \begin{align*} H_{1}\cap \cdots \cap H_{\lambda} \leqslant Z(G). \end{align*} $$
However, by definition,
$Z(G) \leqslant H_{i}\cap H_{j}$
for all
$i,j$
. Consequently, G is a flower group with pistil
$Z(G)$
, which completes the proof.
Example 3.9. Note that the family of
$\mathsf {M}_{\mathcal {C}}$
-groups is not
$\mathsf {MCB}$
(and the same holds for flower groups, since all noncyclic
$\mathsf {M}_{\mathcal {C}}$
-groups are flower groups). As an example, let
$G = G_k = C_3 \rtimes C_{2^k}$
with the action given by inversion. Each
$G_k$
is an
$\mathsf {M}_{\mathcal {C}}$
-group and there is a short exact sequence
$1 \to Z(G_k) \to G_k \to \mathsf {S}_{3} \to 1$
, where
$Z(G_k)$
is a cyclic group of order
$2^{k-1}$
. By Proposition 2.1(b),
$\lambda (G_k)=4$
for every k. In particular, this family is not
$\mathsf {MCB}$
.
3.4 Proof of Theorem 1.3
The next lemma gives a classification of flower groups and expresses the order of these groups in terms of the orders of their petals.
Proposition 3.10 (Qureshi and Reis [Reference Qureshi and Reis9, Proposition 3.3]).
For a finite noncyclic group G and a cyclic subgroup
$C_0$
of G, the following are equivalent:
-
(a) G is a flower group with pistil
$C_0$
; -
(b) for any
$g\in G\setminus C_0$
, there exists a unique maximal cyclic group
$\mathcal {C}$
such that
$g\in C$
and
$C_0\subseteq C$
; -
(c) there exist
$k\geqslant 1$
and distinct maximal cyclic subgroups
$C_1, \ldots , C_k$
of G such that
$C_i\cap C_j=C_0$
for
$1\leqslant i<j\leqslant k$
and
$$ \begin{align*}\sum_{i=1}^k|C_i|-(k-1)|C_0|=|G|.\end{align*} $$
Proof of Theorem 1.3.
(1) Let G be a noncyclic group and let
$H_1,\ldots ,H_{\lambda }$
be the maximal cyclic subgroups of G, where
$\lambda = \lambda (G)$
. By Proposition 2.1(a), the union
$\bigcup _{i=1}^{\lambda } H_i$
forms an irredundant cover of G. By Lemma 2.2,
$$ \begin{align*} |G|\leqslant \sum_{i=1}^{\lambda-1}|H_i|. \end{align*} $$
Since each
$H_i$
is cyclic, its order divides
$\exp (G)$
. Therefore,
$$ \begin{align*} |G|\leqslant \sum_{i=1}^{\lambda-1}|H_i|\leqslant (\lambda-1) \cdot \exp(G), \end{align*} $$
which is the claimed bound.
(2) Assume G is a flower group. As in item (1), the maximal cyclic subgroups form an irredundant cover and these subgroups are precisely the petals
$\mathcal {C}_1,\ldots ,\mathcal {C}_{\lambda }$
. Order the
$\mathcal {C}_i$
so that
$|\mathcal {C}_i|\geqslant |\mathcal {C}_{i+1}|$
for each i. With this ordering, it suffices to prove that
$$ \begin{align*} |\mathcal{C}_1| \leqslant (\lambda-1)\cdot |\mathcal{C}_0|. \end{align*} $$
Combining Lemma 2.2 and Proposition 3.10, we obtain
$$ \begin{align*} \sum_{i=1}^{\lambda}|\mathcal{C}_i|-(\lambda-1)|\mathcal{C}_0|=|G| \leqslant \sum_{i=2}^{\lambda}|\mathcal{C}_i|, \end{align*} $$
and the desired inequality for
$|\mathcal {C}_1|$
follows.
It remains to bound the order
$|G|$
. By definition,
$\mathcal {C}_0 = \mathcal {C}_1 \cap \mathcal {C}_2$
. By Lemma 2.3,
$|G:\mathcal {C}_i| \leqslant \lambda -1$
for
$i \in \{1,2\}$
. By the Poincaré lemma [Reference Robinson10, 1.3.11], the index
$$ \begin{align*} |G: \mathcal{C}_0| = |G: \mathcal{C}_1 \cap \mathcal{C}_2| \leqslant |G \colon \mathcal{C}_1| \cdot |G \colon \mathcal{C}_2| \leqslant (\lambda-1)^2.\end{align*} $$
In particular, the order
$|G|$
is at most
$(\lambda -1)^2 \cdot |\mathcal {C}_0|$
, as required.
(3) Let G be a flower p-group with pistil
$\mathcal {C}_0$
. Set
$m= \mathsf {lcm}(|\mathcal {C}_1|, |\mathcal {C}_2|, \ldots , |\mathcal {C}_{\lambda }|)$
. Since G is a p-group, each
$\mathcal {C}_i$
has order a power of p, so that every element order divides m. Thus,
$m=\exp (G)$
. By item (2), the exponent
$\exp (G)$
divides
$|\mathcal {C}_0|\cdot (\lambda -1)$
. This completes the proof.
We finish the paper with two remarks that provide concrete examples illustrating the scope and limitations of Theorem 1.3.
Remark 3.11. The bound in Theorem 1.3(1) is attained for the quaternion group, but for many groups, it is much larger than
$|G|$
. We propose the following refinement. Define
$$ \begin{align*} M := \max\{|H| \mid H \text{ is a maximal cyclic in } G\}. \end{align*} $$
Since the bound in Theorem 1.3(1) involves the order of a maximal cyclic subgroup, the same conclusion remains valid if we replace
$\exp (G)$
with M. In fact, using M gives a sharper estimate in general: for p-groups, we have
$M=\exp (G)$
and for many groups of mixed order, the bound becomes strictly smaller. For example, if
$G = \mathsf {SL}(2, 3)$
, then
$|G|=24$
,
$\exp (G)=12$
,
$|M|=6$
and
$\lambda (G)=7$
. In particular, the resulting bound is
$$ \begin{align*} |G| \leqslant M \cdot (\lambda-1) = 36 < \exp(G) \cdot (\lambda-1) = 72. \end{align*} $$
Remark 3.12. The parameters
$\lambda (G)$
,
$\exp (G)$
and
$|\mathcal {C}_0|$
behave independently in general.
-
(1) The parameters
$\lambda (G)$
and
$\exp (G)$
are independent. If G is an elementary abelian p-group of rank r, then
$\exp (G)=p$
and In contrast, if p is an odd prime and
$$ \begin{align*} \lambda(G)=\frac{p^{r}-1}{p-1}. \end{align*} $$
$ G = C_2 \times C_2 \times C_p$
, then
$\lambda (G)=3$
, while
$\exp (G)=2p$
.
-
(2) Let
$n \geqslant 2$
. If
$G = D_{2^n}$
, then G is a flower group with trivial pistil. Moreover,
$\lambda (D_{2^n}) = 2^n + 1$
and
$\exp (D_{2^n}) = 2^n$
. Thus, the bounds in Theorem 1.3(2) and (3) are attained for this family. -
(3) Let p be an odd prime. If
$G = D_p$
, then
$\lambda (G)=p+1$
and
$\exp (G)=2p$
. This shows that
$\exp (G)$
cannot be bound by
$|\mathcal {C}_0|(\lambda (G)-1)$
without imposing additional assumptions. -
(4) To see the independence of
$\lambda (G)$
and the size of the pistil, take
$n\geqslant 2$
and consider the group
$G =\langle a,b \mid a^3 = 1 = b^{2^n},\ a^{\,b} = a^{-1} \rangle $
. This is a flower group with
$|\mathcal {C}_0| = 2^{n-1}$
and
$\lambda (G) = 4$
.
