c
⃝ T¨UB˙ITAK
doi:10.3906/mat-1704-81
h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t h / Research Article
Some results on Hecke and extended Hecke groups
Recep S¸AH˙IN∗Department of Mathematics, Faculty of Arts and Sciences, Balıkesir University, Balıkesir, Turkey
Received: 19.04.2017 • Accepted/Published Online: 02.06.2017 • Final Version: 24.03.2018
Abstract: Let q≥ 3 be a prime number and let H(λq) be the extended Hecke group associated with q. In this paper, we determine the presentation of the commutator subgroup ( H ( λq)α)′ of the normal subgroup H ( λq)α , where H ( λq)α is a subgroup of index 2 in H ( λq). Next we discuss the commutator subgroup ( H2)′( λq) of the principal congruence subgroup H2( λq) of H ( λq) . Then we show that some quotient groups of H ( λq) are generalized M∗−groups. Finally, we prove some results related to some normal subgroups of H ( λq) , especially in the case q = 5.
Key words: Extended Hecke groups, commutator subgroups, principal congruence subgroups, generalized M∗−groups
1. Introduction
In [14], Hecke introduced the Hecke groups H(λ) generated by two linear fractional transformations
T (z) =−1
z and S(z) =−
1
z + λ,
where λ is a fixed positive real number.
He showed that H( λ ) is Fuchsian if and only if λ = λq = 2 cosπq, for integer q≥ 3, (λ < 2), or λ ≥ 2. We
will focus on the discrete case with λ < 2 and we denote it by H ( λq). The Hecke group H ( λq) is isomorphic
to the free product of two finite cyclic groups of orders 2 and q ,
H(λq) =< T, S | T2= Sq = I >∼= C2∗ Cq, [10].
Let Γ be a subgroup of finite index in H(λq) . Then U/Γ , where U is the upper half plane of a Riemann
surface. Let g and t be the genus and the number of cusps of U/Γ, respectively, and let m1,· · · , mk be the
branching numbers of the branch points on U/Γ. The signature of Γ is (g; m1, ..., mk; t) .
The Hecke group H(λq) can be thought of as triangle groups having an infinity as one of the parameters.
As the signature of H(λq) is (0; 2, q,∞), each is an infinite triangle group. Moreover, the quotient space
U/H(λq) is a sphere with one puncture and two elliptic fixed points of order 2 and q. Hence the surface
U/H(λq) is an orbifold.
Examples of these groups are H(λ3) = Γ = P SL(2,Z) (the modular group), H(λ4) = H(
√
2), H(λ5) =
H(1+2√5), and H(λ6) = H(
√
3). It is clear that H(λq)⊂ P SL(2, Z[λq]) , for q≥ 4.
∗Correspondence: rsahin@balikesir.edu.tr
2010 AMS Mathematics Subject Classification: 20H10, 11F06, 30F50 In memory of my dear son Can S¸ahin
The extended modular group, denoted by H ( λ3) = Π = P GL(2,Z), is defined by adding the reflection
R(z) = 1/ z to the generators of the modular group H ( λ3) . Then the extended Hecke group, denoted by H(λq) ,
has been defined in [35] and [39] similar to the extended modular group by adding the reflection R(z) = 1/z to the generators of the Hecke group H(λq). Thus the extended Hecke group H(λq) has the presentation
H(λq) =< T, S, R| T2= Sq = R2= (T R)2= (RS)2= I >∼= D2∗C2Dq. (1) If we take R1(z) = 1z, R2(z) =−z, R3(z) =−z − λq, where T = R2R1 = R1R2 and S = R1R3; then
we get the alternative presentation
H(λq) =< R1,R2,R3| R21= R 2 2= R
2
3= (R1R2)2= (R1R3)q = I > .
The Hecke group H(λq) is a subgroup of index 2 in H(λq). Since the extended Hecke groups H ( λq)
contain a reflection, they are proper non-Euclidean crystallographic (N EC) groups [28]. Thus the quotient space U/H (λq) is a Klein surface and U/H (λq) is the canonical double cover of U/H (λq) .
The Hecke groups H ( λq) , the extended Hecke groups H ( λq) and their normal subgroups have been
studied for many aspects in the literature (for instances, please see [1,2, 6, 7, 13, 17, 24, 32, 36,and45]). Here the map
α : T → RT, S → S, R → R, (2)
induces an outer automorphism of H(λq) , [17, p. 12]. Thus the group
H(λq)α =< RT, S| (RT )2= Sq = I >∼= C2∗ Cq, (3)
is a subgroup of index 2 in H ( λq).
Throughout this paper, we identify matrix A in GL(2,Z[λq]) with −A, so that they each represent the
same element of H(λq). Thus we can represent the generators of H(λq) as
T = ( 0 −1 1 0 ) , S = ( 0 −1 1 λq ) and R = ( 0 1 1 0 ) .
Next we give some information about the principal congruence subgroups of H(λq).
The principal congruence subgroups Hp(λq) of level p, p prime, of H(λq) are defined in [38] (see
also [15] and [26]) as Hp(λq) = { A = ( a bλq cλq d ) ∈ H(λq) : a≡ d ≡ ±1, b ≡ c ≡ 0 (mod p), det A = ±1 } .
Hp(λq) is always a normal subgroup of finite index in H(λq). It is easily seen that Hp(λq) = Hp(λq)∩ H(λq).
By [38], we know that if p ≥ 3 is a prime number, then Hp(λq) = Hp(λq) and if p = 2, then
H(λq)/H2(λq) ∼= H(λq)/H2(λq). Thus, the groups H2(λq) and H2(λq) are very important.
The principal congruence subgroups H2( λ3) = Γ(2) , H2( λ3) = Π(2) and Γ(4) = Π(4) of Γ and Π ,
respectively, have been studied extensively in the literature, for example, in relation to number theory, modular forms, modular curves, Belyi’s theory, and graph theory (for instance, see [8,11, 12, 21, 22, 33,and40]).
Some normal subgroups (the first and the second commutator subgroups H′(λq) and H′′(λq), the
principal congruence subgroups Hp(λq) and the m−th power subgroups H m
(λq) ) of H(λq) , q ≥ 3 prime
number, have been studied by Ikikardes, Koruo˘glu, Sahin, and Bizim in [38, 42, 43, 44]. For q ≥ 3 a prime number, they proved the following results:
a) There are exactly 3 normal subgroups of index 2 in H(λq). Namely, H(λq) =< T, S | T2 = Sq = I >∼=
C2∗ Cq, H0(λq) =< R, S, T ST | R2 = Sq = (T ST )q = (RS)2 = (RT ST )2 = I >∼= Dq ∗Z2 Dq, and
H(λq)α =< T R, S| (T R)2= Sq = I >∼= C2∗ Cq.
b) There is exactly one normal subgroup of index 4 in H(λq). Namely, H′(λq) =< S, T ST | Sq = (T ST )q =
I >∼= C2∗ Cq .
c) There are exactly 2 normal subgroups of index 2q in H(λq). Namely, Hq(λq) =< T > ∗ < ST Sq−1 >
∗ . . . ∗ < Sq−1T S >∼ = C|2∗ C2{z∗ · · · ∗ C2} q times , and H2(λq) =< T R > ∗ < RST S > ∗ . . . ∗ < RSq−1T Sq−1 >∼= C2∗ C2∗ · · · ∗ C2 | {z } q times .
d) The second commutator subgroup H′′(λq) of H(λq) is a normal subgroup of index 4q2 in H(λq) . Namely
H′′(λq) is a free group with basis [S, T ST ], [S, T S2T ], ..., [S, T Sq−1T ], [S2, T ST ], [S2, T S2T ], ..., [S2, T Sq−1T ],
..., [Sq−1, T ST ], [Sq−1, T S2T ], ..., [Sq−1, T Sq−1T ].
e) The group (H2)′(λ
q) is equal to the second commutator subgroup H′′(λq) and it has index q in H′(λq).
f) The group (Hq)′(λ
q) is a free group of rank 1 + (q− 2)2q−1, of signature ((q− 3)2q−2+ 1; ∞(2
q−1)
) and of index 2q+1q in H(λ
q).
Using the above results , we get the following subgroup diagram in Figure1.
On the other hand, when Sahin et al. were studying in [42] some normal subgroups of the extended Hecke groups H(λq), for q ≥ 3 prime, they came across an interesting general fact. If a bordered surface
group Γ is a normal subgroup of finite index in H(λq), then H(λq)/Γ is a group of automorphisms of the
bordered Klein surface X =U/H(λq). Moreover, the automorphism groups G of order (q4q−2)(g−1) that act on
compact bordered Klein surfaces X of algebraic genus g≥ 2 are finite quotient groups of the extended Hecke groups H(λq), where q ≥ 3 is an integer. For example, the groups of orders |G| = 12(g − 1), |G| = 8(g − 1),
|G| = 20
3(g− 1), respectively, are the finite quotient groups of the extended Hecke groups H(λ3) , H(λ4) , or
H(λ5) [41] and [3]. Here the orders of these groups are the highest three among the automorphism groups of
the compact Klein surfaces of algebraic genus g ≥ 2 (see [31, p. 221, proposition 1]). The groups of order
|G| = 12(g − 1) are M∗−groups. These groups were first introduced in [30], and have been studied in several
papers [4] and [5] .
Sahin et al. defined the generalized M∗−groups in [41] similar to the M∗−groups. A finite group G is called a generalized M∗−group if it is generated by three distinct nontrivial elements r1, r2, r3 that satisfy the
) (λ H q ) H(λq H0(λq) H(λq)α ) '(λ H q =H(λq) 2 ) (λ Hq q H (λ) q 2 ) '(λ H q H2(λq) ) ''(λ H q =(H )'(λq) 2 ) )'(λ (H q q 2 2 2 q 2 2 q q q q 2 2 2 q 2q-1 Figure 1. r21= r22= r32= (r1r2)2= (r1r3)q = I,
q ≥ 3 prime, and other relations that make the group finite. It is clear that the order of G is (q4q−2)(g− 1),
g≥ 2 integer.
In [41], Sahin et al. showed that a finite group of order at least 4q is a generalized M∗−group if and only if it is the homomorphic image of the extended Hecke group H(λq) , q≥ 3 prime. Thus, if the index of a normal
subgroup N in H(λq) is greater than 2q, then the quotient group H(λq)/N is a generalized M∗−group.
The aim of this work is to generalize some results known for the extended modular group to the extended Hecke groups H(λq) , for q≥ 3 prime (especially, H(λ5)). We obtain the group structures and the signatures of
the commutator subgroups ( H ( λq)α)′ and ( H2)′( λq) of H ( λq) . For this, we apply the Reidemeister–Schreier
method, the permutation method, and the Riemann–Hurwitz formula. Next, we discuss some normal subgroups and how they are related under the map α introduced in (2). We also give some relations between some normal subgroups of H ( λq) and a figure that explain the effect of α. Furthermore, we show that some of the quotient
groups of H ( λq) are generalized M∗−groups. Finally, we determine the generators of H4( λ5) and prove that
( H2)′( λ5) = H4( λ5) and H10(λ5)̸= (H2)′(λ5)∩ (H5)′(λ5).
2. Commutator subgroups of H ( λq)α and H2( λq)
In this section, we study the first commutator subgroups ( H ( λq)α)′ and ( H2)′( λq) of H ( λq) , for q ≥ 3 a
prime number.
Theorem 2.1 Let q≥ 3 be a prime number. Then
i) |H(λq)α : (H(λq)α)′| = 2q,
Proof
i) The quotient group H(λq)α/(H(λq)α)′ is the group obtained by adding the relation T RS = ST R to the
relations of H(λq)α in (3). Then
H(λq)α/(H(λq)α)′∼= C2× Cq.
Therefore, we obtain |H(λq)α : (H(λq)α)′| = 2q.
ii) We choose Σ ={I, S, S2, ..., Sq−1, T R, T RS, T RS2, ..., T RSq−1} as a Schreier transversal for (H(λ q)α)′.
According to the Reidemeister–Schreier method (see [29]), we get the generators of (H(λq)α)′ as the
following. I.T R.(T R)−1= I, I.S.(S)−1= I, S.T R.(T RS)−1 = ST RSq−1RT, S.S.(S2)−1= I, S2.T R.(T RS2)−1= S2T RSq−2RT, S2.S.(S3)−1 = I, .. . ... Sq−1.T R.(T RSq−1)−1= Sq−1T RSRT, Sq−1.S.(I)−1= I, T R.T R.(I)−1= I, T R.S.(T RS)−1= I, T RS.T R.(S)−1 = T RST RSq−1, T RS.S.(T RS2)−1 = I, T RS2.T R.(S2)−1= T RS2T RSq−2, T RS2.S.(T RS3)−1= I, .. . ... T RSq−1.T R.(Sq−1)−1= T RSq−1T RS, T RSq−1.S.(T R)−1 = I. Here ST RSq−1RT = ST ST , S2T RSq−2RT = S2T S2T, Sq−1T RSRT = Sq−1T Sq−1T, T RST RSq−1 = T Sq−1T Sq−1, T RS2T RSq−2 = T Sq−2T Sq−2 and T RSq−1T RS = T ST S , as T R = RT and SR = RSq−1.
Also as (ST ST )−1= T Sq−1T Sq−1, (S2T S2T )−1 = T Sq−2T Sq−2 and (Sq−1T Sq−1T )−1= T ST S, the
genera-tors of (H(λq)α)′ are T ST S, T S2T S2, ..., T Sq−1T Sq−1.
Using the permutation method (see [46]) and the Riemann–Hurwitz formula, we get the signature of (H(λq)α)′ as (0;∞, ∞, · · · , ∞| {z }
q times
) = (0; ∞(q)). 2
It is clear that the group (H(λq)α)′ is a subgroup of H(λq). From [27], there are only two normal
subgroups of index 2q in H(λq), for q≥ 3 prime. Namely, H′(λq) and H2(λq). As the signature of H2(λq) is
(0; ∞(q)) (see [16]), we get the following result.
Corollary 2.2 The subgroup (H(λq)α)′ is equal to the principal congruence subgroup H2(λq) of H(λq), i.e.
(H(λq)α)′ = H2(λq).
Theorem 2.3 Let q≥ 3 be a prime number.
i) H2(λq) : (H2)′(λq) = 2q.
ii) The group (H2)′(λq) is a free group of rank 1 + (q− 2)2q−1.
Proof
i) If we take k1= T R, k2= RST S, k3= RS2T S2, · · · , kq = RSq−1T Sq−1 as the generators of H2(λq), then
the quotient group H2(λq)/(H2)′(λq) is the group obtained by adding the relation kikj = kjki to the
relations of H2(λq), for i̸= j and i, j ∈ {1, 2, · · · , q}. Thus we have
H2(λq)/(H2)′(λq) ∼= C|2× C2× · · · × C{z 2} q times
.
Therefore, we obtain H2(λq) : (H2)′(λq) = 2q.
ii) Let Σ = {I, k1, k2, ..., kq, k1k2, k1k3, ..., k1kq, k2k3, k2k4, ..., k2kq, ..., kq−1kq, k1k2k3, k1k2k4,
..., k1k2kq, ..., k1k2· · · kq} be a Schreier transversal for (H2)′(λq) . Using the Reidemeister–Schreier
method, we obtain the generators of (H2)′(λq) as the following.
There are C(q, 2) = (
q
2 )
generators of the form kikjkikj, where i < j and i, j∈ {1, 2, · · · , q} . There are
2× (
q
3 )
generators of the form kikjktkjktki, or kikjktkiktkj, where i < j < t and i, j, t∈ {1, 2, · · · , q} .
There are 3× (
q
4 )
generators of the form kikjktkukikuktkj, or kikjktkukjkuktki, or kikjktkuktkukjki,
where i < j < t < u and i, j, t, u∈ {1, 2, · · · , q} . Similarly, there are (q − 1) × ( q q ) generators of the form k1k2· · · kqk1kqkq−1· · · k2, or k1k2· · · kqk2kqkq−1· · · k3k1, or · · · , or k1k2· · · kqkq−1kqkq−2· · · k2k1.
Totally, there are 1 + (q− 2)2q−1 generators of (H 2)′(λq) .
iii) We know that H(λq) : (H2)′(λq) = 2q.q and|H(λq) : H2(λq)| = 2q . Therefore we get H2(λq) : (H2)′(λq) =
2q−1.
Finally, we find the signature of (H2)′(λq) as (q2q−3− 2q−1+ 1;∞, ∞, · · · , ∞| {z } q.2(q−2)times
) = ((q− 4)2q−3+ 1;
∞(q.2q−2)). 2
Corollary 2.4 We have H2(λq) = (H2)′(λq)(H2)′(λq).
Proof As (H2)′(λ
q) and (H2)′(λq) are normal subgroups of H′(λq), we obtain the chains
(H2)′(λq)⊂ (H2)′(λq)(H2)′(λq)⊂ H2(λq) and (H2)′(λq)⊂ (H2)′(λq)(H2)′(λq)⊂ H2(λq).
Then we get the index H2(λq) : (H2)′(λq)(H2)′(λq) divides both of q and 2q−1. Since (q, 2q−1) = 1 ,
we have H2(λq) : (H2)′(λq)(H2)′(λq) = 1 . Thus we get H2(λq) = (H2)′(λq)(H2)′(λq). 2
Corollary 2.5 We have
a) H(λq) : ((H2)′(λq)∩ (H2)′(λq)) = 2q.q2
Proof a) ( H2)′(λq) and (H2)′(λq) are normal subgroups of H(λq) . By one of the isomorphism theorems of
the groups, we have that
((H2)′(λq)(H2)′(λq))/(H2)′(λq) ∼= (H2)′(λq)/((H2)′(λq)∩ (H2)′(λq)). As (H2)′(λq)(H2)′(λq) ∼= H2(λq), we find H2(λq)/(H2)′(λq) ∼= (H2)′(λq)/((H2)′(λq)∩ (H2)′(λq)). Then H2(λq) : (H2)′(λq) = (H2)′(λq) : ((H2)′(λq)∩ (H2)′(λq)) . As H2(λq) : ((H2(λq))′ = q, we get (H2)′(λq) : ((H2)′(λq)∩ (H2)′(λq)) = q. Thus, we have H(λq) : ((H2(λq))′∩ (H2)′(λq)) =|H(λq) : H2(λq)| . H2(λq) : (H2)′(λq) . (H2)′(λq) : ((H2(λq))′∩ (H2)′(λq)) . As H2(λq) : (H2)′(λq) = 2q−1, we obtain H(λq) : ((H2)′(λq)∩ (H2)′(λq)) = 2q.q2.
b) The proof is similar to a) . 2
Remark 2.6 Under the map α in (2), any subgroup of H(λq) is mapped to a subgroup of H(λq) similar to
the extended modular group in [19] and [20] . Indeed one finds H(λq) α ↔ H(λq)α Hq(λ q) ↔ H2(λq) H′(λq) ↔ H2(λq) (Hq)′(λ q) ↔ (H2)′(λq).
Of course, if we know the generators of any one of these subgroups, then we find the generators of its image under α. The subgroups H(λq), H0(λq), H′(λq) , and H′′(λq) of H(λq) are α−invariant and hence
they are characteristic subgroups. Figure 2summarizes these results .
As shown in [18], if M is a regular or orientably regular hypermap corresponding to a normal subgroup M of H(λq), then Mα is the hypermap corresponding to the normal subgroup Mα.
Corollary 2.7 Let q≥ 3 be a prime number.
i) The quotient groups H(λq)/H′(λq) and H(λq)/H2(λq) are generalized M∗−groups. These quotient groups
act on surfaces of topological type ((q− 1), 1, +) and ((q − 1), q, +) respectively, where in the triple
(g, k, ϵ), g is the algebraic genus, k is the number of boundary components, and ϵ describes the orientability
) (λ H q ) H(λq H0(λq) α α ) H(λq ) '(λ H q =H(λq) 2 ) (λ H q q ) (λ H2 q ) '(λ H q H2(λq)=(H(λq)α)' ) ''(λ H q =(H2)'(λq) ) )'(λ (H q q ) )'(λ H ( 2 q 2 2 2 q 2 2 q q q q 2 2 2 q 2q-1 2q-1
Figure 2. ◦ Normal subgroups of H(λq) • Characteristic subgroups of H(λq).
ii) The quotient group H(λq)/H′′(λq) is a generalized M∗−group. This quotient group acts on surfaces of
topological type (q.(q− 2) + 1, q, +).
iii) The quotient groups H(λq)/(Hq)′(λq) and H(λq)/(H2)′(λq) are generalized M∗−groups. These quotient
groups act on surfaces of topological type (2q−1.(q− 2) + 1, 2q−1, +) and (2q−1.(q− 2) + 1, q.2q−2, +)
respectively.
Remark 2.8 If q = 3, then H(λ3)/H′(λ3) and H(λ3)/H2(λ3) act on surfaces of topological type (2, 1, +) and
(2, 3, +); H(λ3)/H′′(λ3) acts on surfaces of topological type (4, 3, +); H(λ3)/(H3)′(λ3) and H(λ3)/(H2)′(λ3)
act on surfaces of topological type (5, 4, +) and (5, 6, +). If q = 5, then H(λ5)/H′(λ5) and H(λ5)/H2(λ5)
act on surfaces of topological type (4, 1, +) and (4, 5, +); H(λ5)/H′′(λ5) acts on surfaces of topological type
(16, 5, +); H(λ5)/(H5)′(λ5) and H(λ5)/(H2)′(λ5) act on surfaces of topological type (49, 16, +) and (49,
40, +). All these results coincide with some results given in [3].
In the following, we focus on the Hecke group H(λ5). We know from [37] that the subgroup (H2)′(λ3) is
equal to the congruence subgroup H4(λ3). We want to derive a similar equation for H(λ5) . For this we start
with the special example q = 5 in ii) of the proof of Theorem 2.3 .
Example 2.9 Let q = 5 . Then H2(λ5) : (H2)′(λ5) = 32. We choose Σ ={I, k1, k2, k3, k4, k5, k1k2, k1k3,
k1k4, k1k5, k2k3, k2k4, k2k5, k3k4, k3k5, k4k5, k1k2k3, k1k2k4, k1k2k5, k1k3k4, k1k3k5, k1k4k5, k2k3k4,
k2k3k5, k2k4k5, k3k4k5, k1k2k3k4, k1k2k3k5, k1k2k4k5, k1k3k4k5, k2k3k4k5, k1k2k3k4k5} as a Schreier
transversal for (H2)′(λ5) . Using the Reidemeister–Schreier method, we get the following generators of (H2)′(λ5)
(here λ = λ5= 1+ √
5
2 and λ is a root of the polynomial λ
k1k2k1k2= ( 1 4λ 0 1 ) k2k4k2k4= ( 48λ + 29 40λ + 24 −20λ − 12 −16λ − 11 ) k1k3k1k3= ( 16λ + 9 20λ + 12 12λ + 8 16λ + 9 ) k2k5k2k5= ( 4λ + 13 4λ + 8 −4λ − 8 −4λ − 3 ) k1k4k1k4= ( 16λ + 9 12λ + 8 20λ + 12 16λ + 9 ) k3k4k3k4= ( 8λ + 5 8λ + 4 −8λ − 4 −8λ − 3 ) k1k5k1k5= ( 1 0 4λ 1 ) k3k5k3k5= ( 48λ + 29 20λ + 12 −40λ − 24 −16λ − 11 ) k2k3k2k3= ( 4λ + 5 8λ + 4 −4λ −4λ − 3 ) k4k5k4k5= ( 4λ + 5 4λ −8λ − 4 −4λ − 3 )
20 generators of the form,
k1k2k3k1k3k2= ( −24λ − 15 −48λ − 28 −12λ − 8 −24λ − 15 ) k1k2k3k2k3k1= ( 4λ + 5 −8λ − 4 4λ −4λ + 5 ) k1k2k4k1k4k2= ( −42λ − 25 −128λ − 80 −20λ − 12 −52λ − 33) ) k1k2k4k2k4k1= ( 48λ + 29 −40λ − 24 20λ + 12 −16λ − 11 ) k1k2k5k1k5k2= ( −8λ − 3 −20λ − 16 −4λ −8λ − 7 ) k1k2k5k2k5k1= ( 4λ + 13 −4λ − 8 4λ + 8 −4λ − 3 ) k1k3k4k1k4k3= ( −36λ − 23 −72λ − 44 −32λ − 20 −64λ − 39 ) k1k3k4k3k4k1= ( −4λ − 3 4λ + 2 −4λ − 2 4λ + 1 ) k1k3k5k1k5k3= ( −48λ − 31 −60λ − 36 −40λ − 24 −48λ − 41) ) k1k3k5k3k5k1= ( 48λ + 29 −20λ − 12 40λ + 24 −16λ − 11 ) k1k4k5k1k5k4= ( −24λ − 15 −20λ − 8 −32λ − 20 −28λ − 15) ) k1k4k5k4k5k1= ( 4λ + 5 −4λ 8λ + 4 −4λ − 3 ) k2k3k4k2k4k3= ( −100λ − 63 −112λ − 68 48λ + 28 52λ + 33 ) k2k3k4k3k4k2= ( −16λ − 11 −40λ + 44 8λ + 4 16λ + 13 ) k2k3k5k2k5k3= ( −80λ − 51 −92λ − 60 40λ + 24 48λ + 25 ) k2k3k5k3k5k2= ( −80λ − 51 −228λ − 140 40λ + 24 112λ + 69 ) k2k4k5k2k5k4= ( −20λ − 11 −12λ 8λ + 4 4λ + 1 ) k2k4k5k4k5k2= ( −20λ − 11 −52λ − 32 −8λ − 4 20λ + 13 ) k3k4k5k3k5k4= ( −206λ − 127 −92λ − 56 182λ + 108 80λ + 49 ) k3k4k5k4k5k3= ( −44λ − 27 −52λ − 32 40λ + 20 44λ + 29 )
15 generators of the form,
k1k2k3k4k1k4k3k2= ( 68λ + 41 84λ + 52 32λ + 20 40λ + 25 ) k1k2k3k4k2k4k3k1= ( −100λ − 63 112λ + 68 −48λ − 28 52λ + 33 ) k1k2k3k4k3k4k2k1= ( −16λ − 11 40λ + 20 −8λ − 4 16λ + 13 ) k1k2k3k5k1k5k3k2= ( 80λ + 49 160λ + 100 40λ + 24 80λ + 49) ) k1k2k3k5k2k5k3k1= ( −80λ − 51 92λ + 60 −40λ − 24 48λ + 25) ) k1k2k3k5k3k5k2k1= ( −80λ − 51 128λ + 80 −40λ − 24 64λ + 37 ) k1k2k4k5k1k5k4k2= ( 80λ + 49 200λ + 120 32λ + 20 80λ + 49 ) k1k2k4k5k2k5k4k1= ( −20λ − 11 12λ −8λ − 4 4λ + 1 ) k1k2k4k5k4k5k2k1= ( −20λ − 11 52λ + 32 −8λ − 4 20λ + 13 ) k1k3k4k5k1k5k4k3= ( 120λ + 73 136λ + 84 108λ + 60 120λ + 73 ) k1k3k4k5k3k5k4k1= ( −128λ − 79 92λ + 56 −112λ − 68 80λ + 49 ) k1k3k4k5k4k5k3k1= ( −44λ − 27 52λ + 32 −40λ − 20 44λ + 29 ) k2k3k4k5k2k5k4k3= ( 368λ + 221 604λ + 372 −172λ − 112 −288λ − 179 ) k2k3k4k5k3k5k4k2= ( 232λ + 145 588λ + 360 −112λ − 68 −280λ − 175 ) k2k3k4k5k4k5k3k2= ( 56λ + 33 164λ + 96 −40λ − 20 −76λ − 51 )
and 4 generators of the form k1k2k3k4k5k1k5k4k3k2= ( −192λ − 119 −372λ − 228 −92λ − 56 −176λ − 111 ) k1k2k3k4k5k2k5k4k3k1= ( 368λ + 221 −604λ − 372 172λ + 112 −288λ − 179 ) k1k2k3k4k5k3k5k4k2k1= ( 232λ + 145 −588λ − 360 112λ + 68 −280λ − 175 ) k1k2k3k4k5k4k5k3k2k1= ( 76λ + 53 −164λ − 96 40λ + 20 −76λ − 51 )
Therefore , the subgroup (H2)′(λ5) is a free group of rank 49 and of signature (5; ∞(40)).
Corollary 2.10 The subgroup (H2)′(λ5) of H(λ5) is equal to the congruence subgroup H4(λ5), i.e. (H2)′(λ5) =
H4(λ5).
Proof From Theorem 2.3 , the group (H2)′(λ5) is a normal subgroup of index 160 in H(λ5). Moreover, the
congruence subgroup H4(λ5) is a normal subgroup of index 160 in H(λ5) (see [16] and [23]) . Indeed, there are
4 normal subgroups of index 160 in H(λ5) (see [9]). However, from the previous example, all generators of the
group (H2)′(λ5) are congruent to the± I (mod 4). Thus (H2)′(λ5)⊆ H4(λ5) and we get (H2)′(λ5) = H4(λ5).
2
On the other hand, in [34], Newman and Smart showed that
H6(λ3) = (H2)′(λ3)∩ (H3)′(λ3).
Now we show that this equality is not true for the Hecke group H(λ5).
Corollary 2.11 H10(λ5)̸= (H2)′(λ5)∩ (H5)′(λ5).
Proof Since (H2)′(λ5) ⊂ H′(λ5) and (H5)′(λ5) ⊂ H′(λ5) , we have (H2)′(λ5)∩ (H5)′(λ5) ⊂ H′(λ5). If
H10(λ5) = (H2)′(λ5)∩ (H5)′(λ5), then H10(λ5)⊂ H′(λ5). However, this is impossible since the commutator
subgroup H′(λ5) is not congruence, from [25]. Then we get H10(λ5)̸= (H2)′(λ5)∩ (H5)′(λ5). 2
Finally, we formulate the following conjectures. It seems to us difficult to prove them.
Conjecture 2.12 i) For all q≥ 3 prime, (H2(λq))′ = H4(λq).
ii) For all q≥ 3 prime, H2q(λq)̸= (H2)′(λq)∩ (Hq)′(λq).
Acknowledgments
We would like to thank the referees for their valuable suggestions and comments, which helped to improve the presentation of this paper. This paper was supported by Balıkesir University Research Grant no: 2015/49.
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