SOME NORMAL SUBGROUPS OF THE EXTENDED HECKE GROUPSH(λp)
RECEP SAHIN, SEBAHATTIN IKIKARDES AND ¨OZDEN KORUO ˘GLU
ABSTRACT. We consider the extended Hecke groupsH(λp) generated byT (z) = −1/z, S(z) = −1/(z+λp) andR(z) = 1/ ¯
z with λp= 2 cos(π/p) for p ≥ 3 prime number. In this article, we study the abstract group structure of the extended Hecke groups and the power subgroupsHm(λp) ofH(λp). Then, we give the relations between commutator subgroups and power subgroups and also the information of interest about free nor-mal subgroups of the extended Hecke groups.
1. Introduction. In [7], Hecke introduced the groups H(λ) generated by two linear fractional transformations
T (z) = −1
z and U(z) = z + λ, whereλ is a fixed positive real number. Let S = T U, i.e.,
S(z) = − 1 z + λ.
Hecke showed that H(λ) is discrete if and only if λ = λq = 2 cos(π/q), q ∈ N, q ≥ 3, or λ ≥ 2. We will focus on the discrete with λ < 2, i.e., those withλ = λq,q ≥ 3. These groups have come to be known as the Hecke groups, and we will denoteH(λq) for q ≥ 3. Hecke group H(λq) is isomorphic to the free product of two finite cyclic groups of orders 2 andq, and it has a presentation
H(λq) =T, S | T2=Sq =I ∼=C2∗ Cq.
The first several of these groups are H(λ3) = Γ = P SL(2, Z), the modular group, H(λ4) = H(√2), H(λ5) = H((1 +√5)/2), and H(λ6) =H(√3). It is clear thatH(λq)⊂ P SL(2, Z[λq]), forq ≥ 4.
2000 AMS Mathematics Subject Classification. Primary 20H10, 11F06.
Key words and phrases. Extended Hecke group, power subgroup, commutator
subgroup, free normal subgroup.
Received by the editors on October 27, 2004, and in revised form on January 11, 2005.
Copyright c2006 Rocky Mountain Mathematics Consortium 1033
The Hecke groups H(λq) and their normal subgroups have been extensively studied for many aspects in the literature, see [2, 5, 6]. The Hecke groupH(λ3), the modular groupP SL(2, Z), and its normal subgroups have especially been of great interest in many fields of mathematics, for example number theory, automorphic function theory and group theory, see [12, 15, 16].
The extended modular group, denoted byH(λ3) ==P GL(2, Z), has been defined in [10, 11, 22] by adding the reflectionR(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 [8, 17 19] similar to the extended modular group by adding the reflectionR(z) = 1/¯z to the generators of the Hecke group H(λq). In [17 19], there were studied some normal subgroups of the extended Hecke groups H(λq) (commutator subgroups, even subgroups, principal congruence subgroups, Fuchsian subgroups) and some relations between them. Also, in [20, 21], we investigated the power and free subgroups of the extended modular group H(λ3) and the extended Hecke groupH(λ5) and the relations between power subgroups and commutator subgroups. In this work, we continue our study to which properties of Hecke groupsH(λp),p ≥ 3 prime number, hold for the extended Hecke groups H(λp). Firstly, we give a proof of the fact that the extended Hecke group H(λp) is isomorphic to the free product of two finite dihedral groups of orders 4 and 2p with amalgamation Z2. Secondly, we will
study especially the power subgroups Hm(λp) of H(λp), and we will determine abstract group structure and generators of them. In fact, it is a well-known and important result that the only normal subgroups of the Hecke groupsH(λp) containing torsion areH(λp),H2(λp) and Hp(λp) of indices 1, 2, p respectively, see [5]. Here we show that this nicely generalizes to the extended Hecke groupsH(λp) withp ≥ 3 prime number. We specially discuss H2(λp) and Hp(λp) as they are nicely related toH(λp) and its commutator subgroupH(λp). Then, we give a classification theorem for these power subgroups. Also we discuss free normal subgroups of finite index in the extended Hecke groupsH(λp), p ≥ 3 prime number.
2. Extended Hecke groups H(λp) and their decomposition. Extended Hecke groupH(λp) has a presentation
H(λp) =T, S, R | T2=Sp=R2=I, RT = T R, RS = S−1R or
(2.1) H(λp) =T, S, R | T2=Sp=R2= (T R)2= (SR)2=I. Hecke group H(λp) is a subgroup of index 2 in H(λp), and it has a presentation
H(λp) =T, S | T2=Sp=I ∼=C2∗ Cp.
NowH(λp) has trivial center, and its outer automorphism class group OutH(λp) = AutH(λp)/Inn H(λp) is generated by the automorphism fixingT and inverting S, so the action of H(λp) onH(λp) by conjuga-tion induces as isomorphismH(λp) ∼= AutH(λp), withR corresponding to the required outer automorphism.
The function
α : T −→ RT, S → S, R → R
preserves the relations in (2.1), so it extends to an endomorphism of H(λp); since α2 is the identity, α is an automorphism, which cannot be inner since T ∈ H(λp) H(λp) whereas T α = RT /∈ H(λp). Therefore the outer automorphism class group OutH(λp) = AutH(λp)/Inn H(λp) has order 2, being generated by α.
In terms of (2.1) we have
α : T −→ RT, S → S, R → R so that
α(H(λp)) =RT, S | (RT )2=Sp=I. The groupα(H(λp)) is a subgroup of index 2 inH(λp).
Now we give a theorem about the group structure of the extended Hecke groupH(λp):
Theorem 2.1. The extended Hecke group H(λp) is given directly as a free product of two groupsG1, G2 with amalgamated subgroup Z2, where G1 is the dihedral group D2 and G2 is the dihedral group Dp, that is,H(λp) ∼=D2∗Z2Dp.
Proof. The result follows from a presentation of the extended Hecke groupH(λp) given (2.1):
H(λp) =T, S, R | T2=Sp=R2= (T R)2= (SR)2=I.
Let G1 = T, R | T2 = R2 = (T R)2 = I ∼= D2, and let G2 = S, R | Sp=R2= (SR)2=I ∼=Dp. Then H(λp) isG1∗ G2 with the
identificationR = R.
In G1, the subgroup generated by R is Z2; this is also true in G2. Therefore, the identification induces an isomorphism andH(λp) is a generalized free product with the subgroup M ∼= Z2 amalgamated.
3. Power subgroups of H(λp). Let m be a positive integer. Let us defineHm(λp) to be the subgroup generated by themth-powers of all elements ofH(λp). The subgroupHm(λp) is called themth-power subgroup of H(λp). As fully invariant subgroups, they are normal in H(λp).
From the definition one can easily deduce that
Hm(λ) > Hmk(λ) and that
Power subgroups of the Hecke groups were studied in [3 5, 9, 15]. In [3, 5, 15], they proved that
(3.1) H(λp) :H2(λp)= 2, |H(λp) :Hp(λp)| = p, |H(λp) :H(λp)| = 2p, H(λp) =H2(λp)∩ Hp(λp), H2(λ p) =S T ST , Hp(λp) =T ST Sp−1 · · · Sp−1T S H(λp) =T ST Sp−1 · · · T Sp−1T S,
and H2pm(λp) are free groups. Also, Newman proved in [15] that H(λ
3)⊂ H6(λ3)⊂ H(λ3).
Then, power subgroups of the extended modular group H(λ3) were investigated in [20]. They showed that
H(λ3) :H2(λ3) = 4, H(λ3) :H3(λ3) = 1, H2(λ3) =S T ST , H3(λ3) =H(λ3),
H(λ3) =H2(λ3) =H2(λ3), H6(λ3)⊂ H(λ3)⊂ H(λ3),
andH6m(λ3) are free groups.
Here we discuss power subgroups of the extended Hecke groups H(λp), p. prime. Firstly we find a presentation for the quotient H(λp)/Hm(λp) by adding the relation Xm = I to the presentation of H(λp) given by (2.1). The order of H(λp)/ Hm(λp) gives us the index. We have
(3.2)
H(λp)/ Hm(λp) ∼=T, S, R | T2=Sp=R2= (T R)2= (SR)2=I, Tm=Sm=Rm= (T R)m= (SR)m=· · · = I. Then we use the Reidemeister-Schreier process to find the presenta-tion of the power subgroupsHm(λp).
Theorem 3.1. The normal subgroup H2(λp) is isomorphic to the free product of two finite cyclic groups of order p. Also
H(λp)/H2(λp) ∼=C2× C2,
H(λp) =H2(λp)∪ T H2(λp)∪ RH2(λp)∪ T RH2(λp), and
H2(λp) =S T ST .
The elements of H2(λp) are characterized by the requirement that the sum of the exponents ofT is even.
Proof. By (3.2), we obtainT2=R2=I and S = I from the relations Sp=S2=I.
Then we get
H(λp)/H2(λp) ∼=T, R | T2=R2= (T R)2=I ∼=C2× C2,
and therefore
H(λp) :H2(λp) = 4.
Now we choose {I, T, R, T R} as a Schreier transversal for H2(λp). According to the Reidemeister-Schreier method, we can form all possi-ble products:
I.T.(T )−1=I, I.S.(I)−1=S, I.R.(R)−1 =I,
T.T.(I)−1=I, T.S.(T )−1=T ST, T.R.(T R)−1=I, R.T.(T R)−1=RT RT, R.S.(R)−1=RSR, R.R.(I)−1 =I, T R.T.(R)−1=T RT R, T R.S.(T R)−1=T RSRT, T R.R.(T )−1=I. SinceRT RT = I, T RT R = I, RSR = S−1 andT RSRT = T S−1T = (T ST )−1, the generators areS and T ST . Thus we have
and
H(λp) =H2(λp)∪ T H2(λp)∪ RH2(λp)∪ T RH2(λp).
Theorem 3.2. Let p ≥ 3 be a prime number. Then Hp(λp) = H(λp).
Proof. By (3.2), we findS = T = R = I from the relations R2=Rp=I, Sp= (SR)2= (SR)p=I, T2=Tp=I.
Thus we have
H(λp) :Hp(λp) = 1, that is,
Hp(λp) =H(λp).
We can now obtain a classification of these subgroups:
Theorem 3.3. Let m be a positive integer, and let p ≥ 3 be a prime
number.
i)Hm(λp) =H(λp) if 2 m,
ii)Hm(λp) =H2(λp) if 2| m but 2p m.
Proof. i) If 2 m, then by (3.2), we find S = T = R = I from the relations
R2=Rm=I, Sp=Sm= (SR)2= (SR)m=I, T2=Tm=I. ThusH(λp)/Hp(λp) is trivial and henceHm(λp) =H(λp).
ii) If 2| m but 2p m, then (2, m) = 2. By (3.2), we obtain S = T2=R2=I from the relations
These show that
H(λp)/Hp(λp) ∼=T, R | T2=R2= (T R)2=I ∼=D2,
and
H(λp) :Hp(λp) = 4.
Since H2(λp) is the only normal subgroup of index 4, we have Hm(λp) =H2(λp).
Therefore we have only the subgroups H2pm(λp) left to consider. In this case the above techniques do not say much aboutH2pm(λp).
To discuss H2pm(λp), we first need to consider the commutator subgroups of the extended Hecke groupsH(λp):
Theorem 3.4. (i) H(λp)/ H(λp) ∼=V4∼=C2× C2;
(ii)H(λp) =S, T ST | Sp= (T ST )p=I ∼=Cp Cp;
(iii) H(λp)/H(λp) ∼=Vp2, here Vp2 denotes an elementary abelian group of orderp2;
(iv) H(λp) is a free group with basis [S, T ST ], [S, T S2T ], . . . , [S, T Sp−1T ], [S2, T ST ], [S2, T S2T ], . . . , [S2, T Sp−1T ], . . . , [Sp−1, T ST ], [Sp−1, T S2T ], . . . , [Sp−1, T Sp−1T ].
(v) For n > 2,H(λp) :H(n)(λp) = ∞. Proof. (i) (iv). Please refer to [18, 19].
(v) Taking relations and abelianizing we find that the resulting quotient is infinite. It follows thatH(λp) has infinite index inH(λp). Further, since this has infinite index it follows that the derived series from this point on has infinite index.
Notice that H(λp) is a subgroup of index 2 of the extended Hecke groupsH(λp), consisting of the words inT and S for which T has even exponent-sum.
We can give the following results using the Theorem 3.1 and Theo-rem 3.4:
Corollary 3.5. (i)H(λp) =H(λp)∩ α(H(λp)), (ii)H(λp) is a subgroup of index p in H(λp), (iii) H(λp) is a subgroup of index p in H(λp).
Proof. (i) Both H(λp) and α(H(λp)) have index 2, so H(λp) ∩ α(H(λp)) has index 4, and hence we findH(λp) =H(λp)∩ α(H(λp)).
(ii) (iii) It is easily seen from Theorem 3.4 and by (3.1).
Theorem 3.6. The commutator subgroup H(λp) of H(λp) satisfies H(λp) =H2(λp).
Theorem 3.7. Let p ≥ 3 be a prime number.
i)H2(λp) =H2(λp) =H2(λp)∩ Hp(λp); ii) (H(λp))p⊂ H(λp).
Theorem 3.8. Let m be a positive integer. The groups H2pm(λp) are the subgroups of the second commutator subgroupH(λp).
Proof. i) Since H2p(λp) ⊂ (H2(λp))p ⊂ H2(λp) and H(λp) = H2(λp) implies that H2p(λp)⊂ (H(λp))p ⊂ H(λp) and H2pm(λp)⊂ H2p(λp) ⊂ H(λp). Since H(λp) does not contain any reflec-tion, H2pm(λp) does not contain any reflection. Also we know that H2pm(λp)⊂ H2pm(λp). Thus, we get
By means of these results, we are going to be able to investigate the subgroupsH2pm(λp). We have by Schreier’s theorem the following theorem:
Theorem 3.9. The subgroups H2pm(λp) are free. Therefore, H(λp) :H2pm(λp) =H(λp) :H2pm(λp) =H(λp) :H(λp).H(λp) :H2pm(λp) = 2H(λp) :H2pm(λp) sinceH(λp) :H(λp)= 2.
Now we find the subgroups of low index of the extended Hecke groups H(λp). This will be done using the commutator subgroupH(λp) of H(λp).
Lemma 3.10. There are exactly 3 normal subgroups of index 2 in
H(λp). Explicitly these areH(λp) = T, S | T2 =Sp =I ∼=C2∗ Cp, H0(λp) = R, S, T ST | R2 = Sp = (T ST )p = (RS)2 = (RT ST )2 = I ∼=Dp∗Z2Dp and α(H(λp)) =T R, S | (T R)2=Sp=I ∼=C2∗ Cp. Proof. Let N H(λp) with H(λp) :N = 2. Since H(λp)/N is abelian, we haveH(λp)⊃ N ⊃ H(λp).
Now H(λp)/H(λp) = C2× C2 = D2, a Klein 4−group. This has
exactly 3 normal subgroups of index 2. Therefore these pull back to exactly 3 normal subgroups of index 2 in H(λp) containing H(λp). SinceN contains H(λp),N must be one of these.
Notice that these results coincide with the ones given in [1, p. 343, Proposition 3.5] for M∗-groups when q = 3, i.e., G1 = H0(λ3), G2=H(λ3),G3=α(H(λ3)) andG4=H(λ3).
Proof. Suppose N H(λp) withH(λp) :N= 3. LetA = H(λp)/N and so|A| = 3 and thus A is abelian. Therefore N ⊃ H(λp) which is impossible sinceH(λp) :H(λp) = 4.
Notice that Lemmas 3.10 and 3.11 also follow from (2.1) by looking at all possible homomorphisms onto cyclic groups of orders 2 and 3.
Lemma 3.12. There are exactly 2 normal subgroups of index 2p in
H(λp). Explicitly these are Hp(λ
p) =T ST Sp−1 · · · Sp−1T S and
H2(λp) =T R RST S · · · RSp−1T Sp−1.
Notice that H2(λp) is the principal congruence subgroup of the extended Hecke groups H(λp) and H2(λp) is the principal congruence subgroup of Hecke groups H(λp), see [13, 18]. Also α(H2(λp)) = Hp(λ
p).
Therefore, we can form the following diagram which the relation between power subgroups and commutator subgroups of the extended Hecke groupH(λp).
4. Free normal subgroups of H(λp). A well-known theorem of Karrass and Solitar, see [14], states that ifN is a subgroup of G, a free product of two groupsA, B with amalgamated subgroup of U, then N can be obtained by two constructions from the intersection ofN and certain conjugates ofA, B, and U. The constructions are those of a tree product (a special kind of generalized free product) and of a Higman-Neumann-Neumann extension. In the case G is an extended Hecke group,A and B are dihedral groups of order 4 and 2p and U is cyclic of order 2. Thus extended Hecke groupH(λp) has two kinds of normal subgroups: Free ones and free products of some infinite cyclic groups, some cyclic groups of order 2 and orderp, some dihedral groups D2and Dp with some dihedral groups Dm1 and Dm2 with amalgamation Z2
H(λp) H(λp) H0(λp) α(H(λp)) H'(λp) Hp(λp) H2(λp) H'(λp) H2(λp) H''(λp) H2pm(λp)=H2pm(λp) 2 2 2 p 2 2 p p p p 2 2 2 p
FIGURE 1. Some subgroups ofH(λp).
and their group theoretical structures will be important to us. Here we discuss them for primep. These have been done for modular group by Newman in [16] and for Hecke groups H(λq), q prime, by Cang¨ul in [2] and for extended modular groupH(λ3) by Sahin, Ikikardes and Koruo˘glu in [20]. Their results can be generalized to the extended Hecke groups H(λp) with p prime number. When q is a composite odd number, it is possible to obtain similar results, however, it looks difficult to find all normal free subgroups in this case.
Lemma 4.1. Let N be a nontrivial normal subgroup of finite index
inH(λp). ThenN is free if and only if it contains no elements of finite order.
Proof. By (2.1), H(λp) is isomorphic to a free product of D2 and Dp each amalgamated over Z2. A subgroup of finite index in H(λp) is isomorphic to a free product of the groups F , Cr and Dm1∗Z2 Dm2, wherer and each mi divide 2 orp. Thus, if N is a subgroup of finite index inH(λp), it follows that
(4.1) N = F ∗
∗Cr∗
∗(Dm1∗Z2Dm2)
where F is either free or {I} and each Cr is conjugate to {T } or to {S} or to {R} and each Dmi is conjugate to{T, R} or to {S, R}. As N contains no elements of finite order the free product∗Cr∗∗(Dm1∗Z2 Dm2) is vacuous, and also asN is nontrivial, N must be free.
Conversely, ifN is free, then by definition, it contains no elements of finite order.
Notice that this lemma is true for all noncocompact NEC groups.
Lemma 4.2. The only normal subgroups of finite index in H(λp) containing elements of finite order are
H(λp), H(λp), H0(λp), α(H(λp)), H2(λp), Hp(λp) andH2(λp). Proof. LetN be a normal subgroup of finite index in H(λp) containing an element of finite order. ThenN contains an element of order 2 or an element of order p or two elements of order 2 or two elements of order 2 andp or three elements so that two elements of order 2 and an element of order p. From [22], we know that an element of order 2 in Γ is conjugate toT or to R or to T R and an element of order p in Γ is conjugate toS or to S2, . . . , or to S(p−1)/2. Therefore if a normal subgroupN contains an element of finite order, then it contains T or R or T R or S, S2, . . . , S(p−1)/2. Therefore there are nine cases:
(i) If N contains T, R and one of S, S2, . . . , S(p−1)/2, then N = H(λp).
(ii) If N contains T and one of S, S2, . . . , S(p−1)/2, but not R and T R, then H(λp)/N ∼= r | r2 = I ∼= C2, as we have the relations t2 =sp =r2 = (rs)2 = (tr)2 =si =t = I where 1 ≤ i ≤ (p − 1)/2. Then by the Reidemeister-Schreier method we haveN = H(λp).
(iii) If N contains R and one of S, S2, . . . , S(p−1)/2, but not T and T R, then H(λp)/N ∼= t | t2 = I ∼= C2, as we have the relations t2 =sp =r2 = (rs)2 = (tr)2 =si =r = I where 1 ≤ i ≤ (p − 1)/2. Then by the Reidemeister-Schreier method we haveN = H0(λp).
(iv) IfN contains T R and one of S, S2, . . . , S(p−1)/2, but notT and R, thenH(λp)/N ∼=t | t2=I ∼=C2, as, this time, we have the relations
t2=sp =r2= (rs)2 = (tr)2=si = (tr) = I where 1 ≤ i ≤ (p − 1)/2. SimilarlyN = H(λp)α.
(v) If N contains T and R but not S, S2, . . . , S(p−1)/2, then H(λp)/N ∼= C1. Therefore we get N = H(λp). Thus this case is impossible.
(vi) IfN contains one of S, S2, . . . , S(p−1)/2 but not T and R, then H(λp)/N ∼= t, r | t2 = r2 = (tr)2 = I ∼= D2. Then by the
Reidemeister-Schreier method we haveN = H2(λp).
(vii) If N contains T but not R, S, S2, . . . , S(p−1)/2, thenH(λp)/N ∼
= s, r | sp = r2 = (sr)2 = I ∼= Dp as we have the relations t2 = sp = r2 = (rs)2 = (tr)2 = t = I. Then by the
Reidemeister-Schreier method we haveN = Hp(λp).
(viii) If N contains T R but not T , R and S, S2, . . . , S(p−1)/2, then H(λp)/N ∼=t, s | t2=sp = (ts)2 =I ∼=Dp as we have the relations t2 =sp =r2 = (rs)2 = (tr)2= (tr) = I. Then by the
Reidemeister-Schreier method we haveN = H2(λp).
(ix) IfN contains R but not T , S, S2, . . . , S(p−1)/2, then H(λp)/N ∼
=t | t2=I ∼=C2. Therefore we have N = H0(λp). Thus this case is impossible.
Theorem 4.3. Let N be a nontrivial normal subgroup of finite index
in H(λp) different from H(λp), H(λp), H0(λp), α(H(λp)), H2(λp), Hp(λ
p) and H2(λp). Then N is a free group. Proof. It is clear by Lemma 4.1 and Lemma 4.2.
Theorem 4.4. Let N be a normal subgroup of finite index in H(λp) different fromH(λp),H(λp),H0(λp), α(H(λp)),H2(λp), Hp(λp) and H2(λp) such that H(λp) :N=μ < ∞. Then μ is divisible by 4p.
Proof. The quotient group contains subgroups of orders 2, 4 and 2p, so its order is divisible by 4p.
Acknowledgments. We would like to thank the referee for
sugges-tions which have improved the exposition of the results in this paper. REFERENCES
1. Bujalance, E.F.-J. Cirre and P. Turbek, Automorphism criteria for M∗-groups,
Proc. Edinburgh Math. Soc. (2)47 (2004), 339 351.
2. I.N. Cang¨. ul, Normal subgroups of Hecke groups, Ph.D. Thesis, Southampton University, 1993.
3..I. N. Cang¨ul and O. Bizim, Commutator subgroups of Hecke groups, Bull. Inst. Math. Acad. Sinica30 (2002), 253 259.
4..I. N. Cang¨ul, R. Sahin, S. Ikikardes and ¨O. Koruo˘glu, Power subgroups of some
Hecke groups II, Houston J. Math., to appear.
5..I. N. Cang¨ul and D. Singerman, Normal subgroups of Hecke groups and regular
maps, Math. Proc. Cambridge Philos. Soc.123 (1998), 59 74.
6. M. Conder and P. Dobcs´anyi, Normal subgroups of low index in the modular
group and other Hecke groups, Univ. of Auckland Math. Dept. Research Report
Ser.496 (2003), 23 pp.
7. E. Hecke, ¨Uber die bestimmung dirichletscher reichen durch ihre funktional-gleichungen, Math. Ann.112 (1936), 664 699.
8. S. Huang, Generalized Hecke groups and Hecke polygons, Ann. Acad. Sci. Fenn.
Math.24 (1999), 187 214.
9. S. Ikikardes, ¨O. Koruo˘glu and R. Sahin, Power subgroups of some Hecke groups, Rocky Mountain J. Math.36 (2006), 497 508.
10. G.A. Jones and J.S. Thornton, Automorphisms and congruence subgroups of
the extended modular group, J. London Math. Soc.34 (1986), 26 40.
11. R.S. Kulkarni, An arithmetic-geometric method in the study of the subgroups
of the modular group, Amer. J. Math.113 (1991), 1053 1133.
12. M.L. Lang, Normalisers of subgroups of the modular group, J. Algebra 248
(2002), 202 218.
13. M. L. Lang, C.H. Lim and S.P. Tan, Principal congruence subgroups of the
14. W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Dover
Publ., Inc., New York, 1976.
15. M. Newman, The structure of some subgroups of the modular group, Illinois
J. Math.6 (1962), 480 487.
16. , Free subgroups and normal subgroups of the modular group, Illinois J. Math.8 (1964), 262 265.
17. N.Y. ¨Ozg¨ur and R. Sahin, On the extended Hecke groupsH(λq), Turkish J. Math.27 (2003), 473 480.
18. R. Sahin and O. Bizim, Some subgroups of the extended Hecke groups H(λq), Acta Math. Sci.23 (2003), 497 502.
19. R. Sahin, O. Bizim andI. N. Cang¨. ul, Commutator subgroups of the extended
Hecke groups, Czech. Math. J.54 (2004), 253 259.
20. R. Sahin, S. Ikikardes and ¨O. Koruo˘glu, On the power subgroups of the
extended modular group Γ, Turkish J. Math.29 (2004), 143 151.
21. R. Sahin, ¨O. Koruo˘glu and S. Ikikardes, On the extended Hecke groupsH(λ5), Algebra Colloq.13 (2006), 17 23.
22. D. Singerman, PSL(2,q) as an image of the extended modular group with
applications to group actions on surfaces, Proc. Edinburgh Math. Soc. (2)30 (1987),
143 151.
Balıkesir ¨Universitesi, Fen-Edebiyat Fak¨ultesi, M atematik B ¨ol¨um¨u, 10145 Balıkesir, Turkey
E-mail address: rsahin@balikesir.edu.tr
Balıkesir ¨Universitesi, Fen-Edebiyat Fak¨ultesi, M atematik B ¨ol¨um¨u, 10145 Balıkesir, Turkey
E-mail address: skardes@balikesir.edu.tr
BalıkesirUniversitesi, Necatibey Egitim Fak¨¨ ultesi, Ilkogretim B ¨ol¨um¨u, M atematik Egitimi, 10100 Balıkesir, Turkey
