On the some normal subgroups of the extended modular group

On the some normal subgroups of the extended modular group

Recep Sahin

Balıkesir Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, 10145 Çag˘ısß Kampüsü, Balıkesir, Turkey

a r t i c l e

i n f o

Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary

Keywords:

Extended modular group Principal congruence subgroup Commutator subgroup

a b s t r a c t

In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular groupPcontaining the principal congruence subgroupC(12).

Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction

The modular groupCis the discrete subgroup of PSLð2; RÞ generated by two linear fractional transformations tðzÞ ¼ 1

z and sðzÞ ¼  1 z þ 1: Then modular groupChas a presentation

C

¼< t; sjt2¼ s3¼ I >ffi C2 C3:

The signature ofCis ð0; þ; ½2; 3; 1; fðÞgÞ. Also the quotient spaceU=CwhereU is the upper half plane, is a sphere with one puncture and two elliptic fixed points of order 2 and 3. Hence the surfaceU=Cis a Riemann surface.

The extended modular groupPhas been defined by adding the reflection rðzÞ ¼ 1=z to the generators of the modular groupC. The extended modular groupPhas a presentation, see[5],

P

¼< t; s; rjt2¼ s3¼ r2¼ I; rt ¼ tr; rs ¼ s1_{r >;}

or, equivalently,

P

¼< t; s; rjt2¼ s3¼ r2¼ ðrtÞ2¼ ðrsÞ2¼ I >ffi D2Z2D3:

Here t, s and r have matrix representations 0 1 1 0   ; 0 1 1 1   and 0 1 1 0   ;

respectively (in this work, we identify each matrix A in GLð2; ZÞ with A, so that they each represent the same element of PGLð2; ZÞÞ . Thus the modular groupC¼ PSLð2; ZÞ is a subgroup of index 2 in the extended modular groupP.

0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.074

E-mail address:rsahin@balikesir.edu.tr

Contents lists available atScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

The signature of the extended modular groupPis ð0; þ; ½; f2; 3; 1gÞ. Since the extended modular groupPcontain a reflection, it is a non-Euclidean crystallographic (NEC) group, which is a discrete subgroupPof the group PGLð2; RÞ of isom-etries ofU such that the quotient U=Pis a Klein surface. AlsoU=Cis the canonical double cover ofU=P.

The extended modular group and its normal subgroups has been studied for many aspects in the literature, for example, automorphism groups of compact Klein surfaces, number theory, Belyi’s theory, graph theory, regular maps.see[1–3,6,7]. Now we give the followings from[5]. Let A ¼ a b

c d

 

represent a typical element ofP. For each integer n P 1, we define

P

ðnÞ ¼ fA 2

P

ja  d  1 and b  c  0ðnÞg;

C

ðnÞ ¼

P

ðnÞ \

C

:

These are normal subgroups of finite index inP, and called the principal congruence subgroups. If n > 2 thenP(n) =C(n) and if n = 2 thenP(2) PC(2) PP(4) =C(4).

Also the function

a

:t ! rt; s ! s; r ! r

is an automorphism ofPwhich is not inner. The effect of

a

on congruence subgroups of small index is shown in[5, p. 32]and [4, p. 135].

In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular group PcontainingC(12), given in[5]and[4]. We achieve this by applying standart techniques of combinatorial group theory (the Reidemeister–Schreier method and the permutation method).

2. Supergroups ofC(12)

Theorem 2.1

(i) There are exactly 3 normal subgroups of index 2 inPcontaining the principal congruence subgroupC(12). Explicitly these are

C

¼ ht; sjt2_{¼ s}3_{¼ Ii ffi C} 2 C3;

P

0¼ hr; s; tstjr2¼ s3¼ ðtstÞ3¼ ðrsÞ2¼ ðrtstÞ2¼ Ii ffi D3Z2D3;

C

a

¼ htr; sjðtrÞ2¼ s3_{¼ Ii ffi C} 2 C3; where tr ¼ 1 0 0 1   and tst ¼ 1 1 1 0   .

(ii) There is only a normal subgroup of index 4 inPcontaining the principal congruence subgroupC(12). Explicitly this is

P

0¼ hs; tstjs3¼ ðtstÞ3¼ Ii ffi C3 C3:

(iii) There are exactly 2 normal subgroups of index 6 inPcontaining the principal congruence subgroupC(12). Explicitly these are

P

ð2Þa¼ ht; sts2_;s2_tsjt2_{¼ ðsts}2_Þ2 ¼ ðs2_tsÞ2 ¼ Ii ffi C2 C2 C2;

P

ð2Þ ¼ htr; rsts; rs2_ts2_jðtrÞ2 ¼ ðrstsÞ2¼ ðrs2_ts2_Þ2 ¼ Ii ffi C2 C2 C2; where sts2_¼ 1 1 2 1   ; s2_{ts ¼} 1 2 1 1   ; rsts ¼ 1 2 0 1   and rs2_ts2_¼ 1 0 2 1   .

(iv) There are exactly 2 normal subgroups of index 12 inPcontaining the principal congruence subgroupC(12). Explicitly these are

C

0 ¼ htsts2_;ts2_tsi;

C

ð2Þ ¼ htsts; ts2_ts2_i; where tsts2_¼ 2 1 1 1   ; ts2_{ts ¼} 1 1 1 2   ; tsts ¼ 1 2 0 1   and ts2_ts2_¼ 1 0 2 1   .

Using the permutation method and Riemann Hurwitz formula, we obtain the signatures ofP0,C

a

,P0,P(2)

a

,P(2),C0

and C(2) as ð0; þ; ½; fð2; 2; 3; 3ÞgÞ; ð0; þ; ½3; fð2; 2ÞgÞ; ð0; þ; ½3; 3; 1; fðÞgÞ, ð0; þ; ½2; 2; 2; 1; fðÞgÞ; ð0; þ; ½; fð2; 2; 2ÞgÞ; ð1; þ; ½1; fðÞgÞ and ð0; þ; ½1; 1; 1; fðÞgÞ, respectively.

Notice thatP(2)

a

=C3whereC3is the power subgroup ofC(see[9]). Theorem 2.2

(i) jC:Cð3Þj ¼ 12

(ii) The groupC(3) is a free group of rank 3 with basis tststs, ts2ts2ts2and tsts2ts2tst. Proof

(i) The quotient groupC/C(3) has the presentation

C

=

C

ð3Þ ffi ht; sjt2_{¼ s}3_{¼ ðtsÞ}3

¼ Ii ffi A4:

Then, we obtain jC:Cð3Þj ¼ 12.

(ii) Now we chooseR= {I, t, s, s2_{, ts, ts}2_{, tst, ts}2_{t, tsts, ts}2_{ts, tsts}2_{, ts}2_ts2_{} as a Schreier transversal for}

C(3). According to the Reidemeister–Schreier method, we can form all possible products:

I:t:ðtÞ1_{¼ I; I:s:ðsÞ}1_{¼ I;} t:t:ðIÞ1¼ I; t:s:ðtsÞ1¼ I; s:t:ðts2_ts2_Þ1 ¼ ststst; s:s:ðs2_Þ1 ¼ I; s2_:t:ðtstsÞ1 ¼ s2_ts2_ts2_{t; s}2_:s:ðIÞ1 ¼ I; ts:t:ðtstÞ1 ¼ I; ts:s:ðts2_Þ1 ¼ I; ts2_:t:ðts2_tÞ1 ¼ I; ts2_:s:ðtÞ1 ¼ I; tst:t:ðtsÞ1 ¼ I; tst:s:ðtstsÞ1 ¼ I; ts2_t:t:ðts2_Þ1 ¼ I; ts2_t:s:ðts2_tsÞ1 ¼ I; tsts:t:ðs2_Þ1 ¼ tststs; tsts:s:ðtsts2_Þ1 ¼ I; ts2_ts:t:ðtsts2 Þ1¼ ts2tststs2_{t; ts}2_ts:s:ðts2_ts2 Þ1¼ I; tsts2_:t:ðts2_tsÞ1 ¼ tsts2_ts2_{tst; tsts}2_:s:ðtstÞ1 ¼ I; ts2_ts2_:t:ðsÞ1 ¼ ts2_ts2_ts2_{; ts}2_ts2_:s:ðts2_tÞ1 ¼ I:

Since (ststst)1_{= ts}2_ts2_ts2_{, (s}2_ts2_ts2_t)1_{= tststs, and (ts}2_tststs2_t)1_{= tsts}2_ts2_{tst, the generators ofC(3) are b}

1= tststs, b2= ts2ts2ts2 and b3= tsts2ts2tst. Here b1¼ 1 3 0 1   ; b2¼ 1 0 3 1   and b3¼ 4 3 3 2  

:Also, using the permutation method, we get also the signature ofC(3) as ð0; þ; ½1ð4Þ_{; fðÞgÞ. h}

Corollary 2.3. There are exactly 2 normal subgroups of index 24 inPcontaining the principal congruence subgroupC(12). Explic-itly these are

C

ð3Þ ¼

P

ð3Þ ¼ hb1;b2;b3i;

P

ð3Þa¼ hb1

a;

b2

a;

b3

ai;

where b1

a

¼ ts2tsts2r ¼ 1 2 2 3   ; b₂

a

¼ tsts2_{tsr ¼} 3 2 2 1   and b3

a

¼ ts2ts2tststr ¼ 1 2 2 5  

: Also using the permutation method, we get also the signature ofP(3)

a

as ð0; þ; ½; fð1ð4Þ_ÞgÞ.

Notice that the normal subgroups ofPdifferent fromP,P0,C

a

,P(2) andP(3)

a

, does not contain any reflection.

Theorem 2.4 (i) jC2:ðC2Þ0j ¼ 9.

(ii) The group (C2₎0_{is a free group of rank 4 with basis ststs}2_ts2_{t, sts}2_ts2_{tst, s}2_tststs2_{t, s}2_ts2_{tstst and of index 3 inC}0_.

(iii) jC3_:ðC3_Þ0_{j ¼ 8.}

(iv) The group (C3)0_{is a free group of rank 5 with basis tsts}2

tsts2, ts2tsts2ts, tstststs2ts2ts2, ststs2tsts, tststs2tstst and of index 4 in C0_.

Proof. (i) and (ii) SinceP0_=C2_{, we have (}

C2)0_=P0 0

. Then it is easy to see from[5, p. 28]. Also, using the permutation method, we get also the signature of ðC2

Þ0as ð1; þ; ½1; 1; 1; fðÞgÞ ¼ ð1; þ; ½1ð3Þ_{; fðÞgÞ.}

(iii) It is well known thatC3_{has the presentation}

ht; sts2;s2_tsjt2

¼ ðsts2Þ2¼ ðs2tsÞ2¼ Ii ffi C2 C2 C2:

The quotient groupC3/(C3)0_{is the group obtained by adding the abelianizing to the relations ofC}3

. Then

C

3₌

ð

C

3

Þ0ffi C2 C2 C2:

Therefore, we obtain jC3:ðC3Þ0j ¼ 8.

(iv) Now we choose R= {I, t, sts2, s2ts, tsts2, ts2ts, ststs, tststs} as a Schreier transversal for (C3)0_{. According to the}

Reidemeister–Schreier method, we can form all possible products: I:t:ðtÞ1¼ I; tsts2_:sts2_:ðtÞ1 ¼ I; t:t:ðIÞ1 ¼ I; ts2_ts:sts2_:ðtststsÞ1 ¼ ts2_ts2_tsts2_ts2_t; sts2_:t:ðtsts2 Þ1¼ sts2tsts2_{t; ststs:sts}2_: ðs2_tsÞ1 ¼ ststs2tsts; s2_ts:t:ðts2_tsÞ1 ¼ s2_tsts2_{tst; tststs:sts}2_:ðts2_tsÞ1 ¼ tststs2_tstst; tsts2_:t:ðsts2_Þ1 ¼ tsts2_tsts2_{; I:s}2_ts:ðs2_tsÞ1 ¼ I; ts2_ts:t:ðs2_tsÞ1 ¼ ts2_tsts2_{ts; t:s}2_ts:ðts2_tsÞ1 ¼ I; ststs:t:ðtststsÞ1 ¼ stststs2_ts2_ts2_{t; sts}2_:s2_ts:ðststsÞ1 ¼ I; tststs:t:ðststsÞ1¼ tstststs2_ts2_ts2_{; s}2_ts:s2_ts:ðIÞ1 ¼ I; I:sts2_:ðsts2_Þ1 ¼ I; tsts2_:s2_{ts:ðtststsÞ}1 ¼ I; t:sts2_:ðtsts2_Þ1_{¼ I; ts}2_ts:s2_ts:ðtÞ1_{¼ I;} sts2_:sts2_:ðIÞ1 ¼ I; ststs:s2_ts:ðsts2_Þ1 ¼ I; s2_ts:sts2_: ðststsÞ1¼ s2_ts2_tsts2_ts2_{; tststs:s}2_ts:ðtsts2 Þ1¼ I: Since ðsts2_tsts2_tÞ1 ¼ tsts2_tsts2_{; ðs}2_tsts2_tstÞ1 ¼ ts2_tsts2_{ts; tststs}2_ts2_ts2_tÞ1 ¼ tstststs2_ts2_ts2_{; ðs}2_ts2_tsts2_ts2_Þ1 ¼ ststs2_{tsts and} ðts2_ts2_tsts2_ts2_tÞ1

¼ tststs2_{tstst, the generators are d}

1¼ tsts2tsts2, d2¼ ts2tsts2ts; d3¼ tstststs2ts2ts2; d4¼ ststs2tsts and d5¼ tststs2tstst. Here d1¼ 5₃ 3₂   ; d2¼ 2₃ 3₅   ; d3¼ 10₃ 3₁   ; d4¼ 1₃ 3₈   and d5¼ 8₃ 3_1   : Also, since jC:ðC3_Þ0_{j¼ 24 and jC:C0j ¼ 6, we obtain jC0 : ðC}3_{Þ0j ¼ 4. Also, using the permutation method, we obtain the signature of}

(C3₎0_{as ð1; þ; ½1}ð4Þ_{; fðÞgÞ.}

Notice that this theorem was proved by Newman and Smart in[8]. But they did not give the generators of the (C2)0_{and (C}3₎0_.

Also this theorem generalized to the Hecke groups H(kq), q P 3 prime, by Sahin and Koruog˘lu in[12].

Corollary 2.5

(i)P(4)

a

= (C3)0_{and (P(2))}0_=P(4).

(ii) There are exactly 2 normal subgroups of index 48 inP. Explicitly these are

P

ð4Þa¼ hd1;d2;d3;d4;d5i

P

ð4Þ ¼ hd1

a;

d2

a;

d3

a;

d4

a;

d5

ai

where d1

a

¼ ts2ts2ts2ts2¼ 1₄ 0₁   ; d2

a

¼ tstststs ¼ 1₀ 4₁   ;d3

a

¼ ts2tsts2ts2tsts2¼ ₁₂7 4₇   ; d4

a

¼ sts2ts2ts2ts ¼ 3 4 4 5   and d5

a

¼ ts2tstststs2t ¼ 3₄ 4 5  

:Also the signature ofP(4) is ð0; þ; ½1ð6Þ_{; fðÞgÞ.}

Therefore we have only left the subgroupsC(6),C (6)

a

andC(12) to consider. In these cases we can say only the following.

Theorem 2.6

(i) The groupC(6) is a free group of rank 13 and of index 144 inP. The signature ofC(6) is ð1; þ; ½1ð12Þ_{; fðÞgÞ.}

Corollary 2.7. There are exactly 2 normal subgroups of index 144 inP. Explicitly these areC(6) andC(6)

a

.There is exactly a normal subgroup of index 576 inP. This isC(12).

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