The commutator subgroup and the index formula of the Hecke group H 5
|
|
- Blaise Gregory
- 6 years ago
- Views:
Transcription
1 J. Group Theory 18 (2015), DOI /jgth de Gruyter 2015 The commutator subgroup and the index formula of the Hecke group H 5 Cheng Lien Lang and Mong Lung Lang Communicated by James Howie Abstract. Let A be an ideal of ZŒ2 cos.=5/ and let H.A/ be the principal congruence subgroup of level A of the Hecke group H 5. The present article gives explicit formulas for ŒH 5 W H.A/. As a byproduct, we prove that the commutator subgroup of H 5 is not congruence. Consequently, the fact that the commutator subgroup of the modular group is congruence cannot be carried over to the Hecke groups. 1 Introduction 1.1 Hecke groups Let q 3 be a fixed integer. The (homogeneous) Hecke group H q is defined to be the maximal discrete subgroup of SL.2; R/ generated by S and T, where S D ; T D 1 q (1.1) 1 0 with q D 2 cos.=q/. Let A be an ideal of ZŒ q. We define three congruence subgroups of H q as follows: H 0.A/ D ¹.a ij / 2 H q W a 21 2 Aº; (1.2) H 1.A/ D ¹.a ij / 2 H q W a 11 1; a 22 1; a 21 2 Aº; (1.3) H.A/ D ¹.a ij / 2 H q W a 11 1; a 22 1; a 12 ; a 21 2 Aº: (1.4) Let Z 2 D h I i. The (inhomogeneous) Hecke group and its congruence subgroups are defined as G q D H q =Z 2 ; G 0.A/ D H 0.A/=Z 2 ; G 1.A/ D H 1.A/Z 2 =Z 2 ; G.A/ D H.A/Z 2 =Z 2 : The present article determines the index ŒH 5 W H.A/. Note that ZŒ 5 is a principal ideal domain and that the minimal polynomial of 5 over Z is x 2 x 1.
2 76 C. L. Lang and M. L. Lang 1.2 The obstruction Observe that H.A/ is the kernel of the homomorphism f W H 5 SL.2; ZŒ =A/. In the special case A D.2/, it is known that f W H 5 SL.2; ZŒ =2/ Š A 5 and that ŒH 5 W H.2/ D 10 (see [9]). Consequently, the map f is not always surjective and the finding of the index ŒH q W H./ is more involved. As a matter of fact, one will see in Section 6 that f is surjective if and only if the norm of A is prime to 6. Note that our result is in line with the modular group case only if gcd.n.a/; 6/ D 1 (see (1.5)). 1.3 The plan and the main results We consider the decomposition A D Q A i, where N.A i / is a power of some rational prime p i and gcd.n.a i /; N.A j // D 1. To determine the index, we will prove in the following sections that (i) ŒH 5 W H.A/ D Q i ŒH 5 W H.A i / (Section 4), (ii) ŒH 5 W H.A i / can be expressed in terms of the norm N.A i / (Section 5). Statement (i) is achieved by studying the normal subgroup H.A i /H.A j /. Statement (ii) can be done by studying the group structure of SL.2; ZŒ =A i /. Our main result shows that if gcd.n./; 6/ D 1, then ŒH 5 W H.2 a 3 b / D I a J b N./ Y 3.1 N.P / 2 /; (1.5) P j./ where the product is over the set of all prime ideals P that divide, I 0 D 1, I 1 D 10, I a D 52 6.a 1/ if a 2, J 0 D 1, J b D b 1/ if b 1. Our proof is elementary but involves some curious construction of matrices (see Sections 3.3 and 3.4). Note that our strategy works for other groups as well (see [5] for elementary matrix groups of P. M. Cohn [3]). 1.4 First application Subgroups of H q are called congruence if they contain some principal congruence subgroups. It is well known that the commutator subgroup of the modular group SL.2; Z/ is congruence of level 12. As an application of the index formula given in (1.5), we prove that (unlike the modular group case) the commutator subgroup of H 5 is not congruence (Section 7). Two subgroups A and B of SL.2; R/ are commensurable with each other if A \ B is of finite index in both A and B. As the commutator subgroups of SL.2; Z/ D H 3 ; H 4 and H 6 are congruence, our study suggests the following conjecture.
3 Commutator subgroup and index formula of the Hecke group H 5 77 Conjecture. The commutator subgroup of H q is congruence if and only if H q is commensurable with the modular group. A weaker version of the above conjecture is: Conjecture. Let q 5 be a prime. Then the commutator subgroup of H q is not congruence. 1.5 Second application Let f be a function (form) defined on H=.n/, where H is the union of the upper half plane and Q [ ¹1º and where.n/ is the principal congruence subgroup of the modular group D PSL.2; Z/. Then P f j is a function (form) of H=, where the sum is over the set of a complete set of coset representatives of =.n//. With the help of our proof of the index formula (1.5), a complete set of coset representatives of G 5 =G./ can be determined. As a consequence, the above construction for functions (forms) of H= can be extended to G 5 as well. Note that the space of forms for G q has positive dimension (Ogg [12, Theorem 3]). 1.6 Organisation of the paper The remainder of this article is organised as follows. In Section 2, we give results related to SL.2; ZŒ q / and ŒH 5 W H.A/ where A is a prime ideal. Section 3 is devoted to the study of some technical lemmas. In particular, we give a set of generators for G.2/ (see Lemma 3.1) which will be used in the determination of the index ŒH.2/ W H.4/. Section 4 investigates the index formula for H.A/. It is shown that the index formula for the principal congruence subgroup (1.4) is multiplicative. Section 5 gives the closed form of the index ŒH 5 W H.A i /. The main result of this article can be found in Section 6. Throughout the sections, we set D 2 cos : (1.6) 5 2 Known results of SL.2; ZŒ q / and ŒH 5 W H.A/ For A an ideal of ZŒ q, we may define the principal congruence subgroup L.A/ of L q D SL.2; ZŒ q / analogously. The formula for the index of the principal congruence subgroup in L q is easily calculated as the modular group case (see [14]): ŒL q W L.A/ D N.A/ 3 Y P ja.1 N.P / 2 /; (2.1)
4 78 C. L. Lang and M. L. Lang where N.A/ denotes the absolute norm of A in ZŒ q and the product is over the set of all prime ideals P that divide A. Let 2 ZŒ q. It follows easily from (2.1) that ŒH. n / W H. nc1 / N./ 3 : (2.2) Recall that D 2 cos =5. Let be a prime ideal of ZŒ. Denoted by p the smallest positive rational prime in. Applying our results in [9], one has (i) ŒH 5 W H./ D 10 if p D 2, (ii) ŒH 5 W H./ D 120 if p D 3 or 5, (iii) ŒH 5 W H./ D.p 1/p.p C 1/ if p D 10k 1, (iv) ŒH 5 W H./ D.p 2 1/p 2.p 2 C 1/ if 3 p D 10k 3. 3 Technical lemmas 3.1 Results of Kulkarni In [4], Kulkarni applied a combination of geometric and arithmetic methods to show that one can produce a set of independent generators in the sense of Rademacher for the congruence subgroups of the modular group, in fact for all subgroups of finite indices. His method can be generalised to all subgroups of finite index of the Hecke groups G q D H q =Z 2, where q is a prime. See [8, Propositions 8 10 and Section 3] for details. By [8, Section 3], we have the following lemma. Lemma 3.1. The principal congruence subgroup G.2/ D H.2/Z 2 =Z 2 has index 10 in G 5 D H 5 =Z 2. A set of generators is given by µ C 2 2 C 2 1 C D ; ; ; : C 2 2 C 2 1 C Reduced form a Observe that a=b is in reduced form if b is a column vector of some 2 H5. For any a; b 2 ZŒ such that the greatest common divisor of a and b is a unit, applying results of Leutbecher ([10, 11]), there exists a unique n 2 Z such that a n =b n is in reduced form. We shall now give an algorithm that enables us to determine the reduced factor n D e.a=b/.
5 Commutator subgroup and index formula of the Hecke group H 5 79 Let a; b 2 ZŒ n ¹0º be given such that the greatest common divisor of a and b is a unit. Then there exists a unique rational integer q such that (i) a D.q/b C r, jbj (ii) 2 < r jbj 2. We call such a division algorithm pseudo-euclidean (see [13] for more details). In terms of matrices, the above can be written as 1 q a D r : (3.1) b b Note that 1 q 2 H 5 : Applying the pseudo-euclidean algorithm repeatedly, one has a D.q 1 /b C r 1 ; b D.q 2 /r 1 C r 2 ; r 1 D.q 2 /r 2 C r 3 ; : r kc1 D.q kc2 /r kc2 C 0: The finiteness of the algorithm is governed by the fact that the set of cusps of H 5 is QŒ [ ¹1º. Note that, in terms of matrices, the above can be written as a D A r kc2 ; (3.2) b 0 where A 2 H 5. It is clear that gcd.a; b/ D gcd.b; r 1 / D D gcd.r kc1 ; r kc2 / D r kc2 is a unit (since a and b are coprime). As is a primitive unit, there exists some e.a=b/ 2 Z[¹0º such that r kc2 D e.a=b/. Multiplying (3.2) by e.a=b/, one has a e.a=b// b e.a=b/ D A 1 : (3.3) 0 Since A 2 H 5 and 1=0 is a reduced form, we conclude that a e.a=b/ =b e.a=b/ is the reduced form of a=b. We state without proof the following lemma ([8, Proposition 6]).
6 80 C. L. Lang and M. L. Lang Lemma 3.2. One has in reduced form. a b c d 2 H5 if and only if ad bc D 1 and a=c, b=d are Lemma 3.3. Let p 2 N be an odd prime and let a D 2 2, c D p 3. For every m 2 Z, S D 1 acm a2 m c 2 H.m/: (3.4) m 1 C acm Proof. The reduced form of 2=p is 2 2 =p 3. By the definition of reduced form, there exist some u; v 2 ZŒ such that X D 22 u p 3 2 H 5 : (3.5) v As a consequence, XT m X 1 2 H 5 for all m 2 Z. Note that the matrix form of XT m X 1 is given by (3.4). 3.3 Some matrices of H 5 By Lemma 3.2, one can show easily that H 5 contains the following matrices which will be used for our study: 1 C 2 2 C 2 C 2 3 C ; ; ; : 2 1 C 2 2 C C 1 4 C Remarks A key proposition (Proposition 4.3) in our study of the index formula requires the existence of D.a ij / 2 H.p/ such that 1 p (mod p 2 ) (equivalently, that a 11 1; a 22 1; a 12 p; a 21 2.p 2 / ZŒ /. The matrices we introduced in Section 3.3 will be used in Section 5 for that purpose. It is perhaps worthwhile to point out that we did not obtain the above matrices simply by chance but rather by a study of the fundamental domains of the subgroups G.p/. See [8] for more detail. 4 The index ŒH 5 W H./ is multiplicative In this section, we investigate the index formula for the principal congruence subgroup (1.4) and prove that the index formula is multiplicative (Lemma 4.1). Note that the index formula for the inhomogeneous Hecke group G q is not multiplicative.
7 Commutator subgroup and index formula of the Hecke group H 5 81 Lemma 4.1. Let, 2 ZŒ and let a and b be the smallest positive rational integers in the ideals./ and./ respectively. Suppose that gcd.a; b/ D 1. Then H 5 D H./H./ and ŒH 5 W H./ D ŒH 5 W H./ ŒH 5 W H./ : Proof. Let K D H./H./. It is clear that 1 a 1 b ; 2 K: (4.1) Since gcd.a; b/ D 1, there exist some m; n 2 Z such that am C bn D 1. As a consequence, we have m n T D 1 a 1 b D 1 2 K: (4.2) Since H 5 is generated by S and T, every element in H 5 is a word w.s; T /. Since T 2 K, it follows that SK D KS (K is normal) and w.s; T /K is S i K for some i (S has order 4 in H 5 ). Hence the index of K in H 5 is either 1, 2 or 4. This implies that.st / 4 2 K. Since the order of ST is 5, we conclude that ST 2 K. It follows that ST; T 2 K. As a consequence, we have S 2 K. Hence K D H 5. The lemma can now be proved by the Second Isomorphism Theorem and the fact that H./ \ H./ D H./. Remark 4.2. In the case q D 5, ZŒ 5 is a principal ideal domain. As a consequence, every ideal of ZŒ 5 takes the form./ for some. Proposition 4.3. Let p 2 N be an odd prime. Suppose that H.p/ contains an element such that.mod p 2 /. Then 1 p ŒH.p n / W H.p nc1 / D p 6 for all n 2 N and ŒH 5 W H.p n / D p 6.n 1/ ŒH 5 W H.p/ : Proof. Let X D pn 1 and A D T pn 2 H.p n /. It follows that X 1 pn.mod p nc1 / and A D 1 pn :
8 82 C. L. Lang and M. L. Lang As a consequence, H.p n / contains A, SAS 1, TSAS 1 T 1, and X, SXS 1 and TSXS 1 T 1. The above matrices modulo p nc1 are given by 1 p n 1 p n 2 p n 3 ; p n ; 1 p n 1 C p n 2 ; 1 p n 1 p n p n 2 ; p n ; 1 p n 1 C p n : Applying Lemma A.1 in Appendix A, they generate an elementary abelian group of order p 6. Hence ŒH.p n / W H.p nc1 / p 6 : By (2.2), ŒH.p n / W H.p nc1 / D p 6. Corollary 4.4. Suppose that H.4/ contains some such that 1 4.mod 8/. Then ŒH.2 n / W H.2 nc1 / D 2 6 for n 2 and ŒH 5 W H.2 n / D 2 6.n 2/ ŒH 5 W H.4/ : Remark. In the case p D 2, H.2/ possesses no such that 1 2.mod 4/. Consequently, Corollary 4.4 cannot be improved to H.2/. 5 The index ŒH 5 W H./ where N./ is a power of a prime Throughout this section, N./ is a power of a rational prime p. There are five cases to consider: (i) p D 5, (ii) p D 2, (iii) p D 3, (iv) p 3 is of the form 10k 3, (v) p is of the form 10k N./ is a power of 5 Let D 2 C. The case p D 5 is slightly different as 5 D 2 2 ramified totally in ZŒ. As a consequence, D n for some n. We shall first determine the order
9 Commutator subgroup and index formula of the Hecke group H 5 83 of H.5 n /=H.5 nc1 /. Lemma 3.3 (p D 5, m D 5) implies that S 2 H.5/, where S.mod 25/: By Proposition 4.3, ŒH.5 m / W H.5 mc1 / D 5 6 if m 1. By (2.2), ŒH. m / W H. mc1 / 5 3. Hence ŒH. m / W H. mc1 / D 5 3 for m 2: In the case m D 1, applying our results in Section 3.3, H. C 2/ contains the following matrix: C 5 a D D T 2 3 C H. C 2/: 4 C C Let J D 1 0. Then J 2 Aut H5. Let D ¹a; b D SaS 1 ; c D JaJ 1 º H./: The matrices in modulo 5 are given by 4 0 a I C. C 2/ b I C. C 2/ c I C. C 2/ 0 4.mod 5/;.mod 5/;.mod 5/: The set generates a group of order 5 3 modulo 5. By (2.2), H./=H.5/ Š h i and jh./=h.5/j D 5 3. In summary, one has ŒH. m / W H. mc1 / D 5 3 for m 1: (5.1) Recall that ŒH 5 W H./ D 120 and that N.x/ is the absolute norm of x. It follows from (5.1) that ŒH 5 W H. m / D m 1/ D N. m / 3.1 N./ 2 / for m 1: (5.2)
10 84 C. L. Lang and M. L. Lang 5.2 N./ is a power of 2 Since 2 is a prime in ZŒ, we have D 2 m. Note first that ŒH 5 W H.2/ D 10. We shall now determine the index ŒG.2/ W G.4/. Applying Lemma 3.1, a set of generators of G.2/ is given by µ C 2 2 C 2 1 C D ; ; ; : C 2 2 C 2 1 C 2 Since 2 generates G.2/, one can show by direct calculation that G.2/=G.4/ is elementary abelian of order 2 4. Since I 2 H.2/ H.4/, one has ŒH.2/ W H.4/ D 2 5 : We now study ŒH.2 n / W H.2 nc1 / for n 2. By (2.2), ŒH.2 n / W H.2 nc1 2 6 : By our results in Section 3.3, C 2 2 C 2 2 H.4/=H.8/: 2 1 C 2 By Corollary 4.4, ŒH 5 W H.2/ D 10; ŒH 5 W H.2 n / D n 1/ for n 2: (5.3) 5.3 N./ is a power of 3 Since 3 2 ZŒ is a prime, D 3 m. Note first that ŒH 3 W H.3/ D 120. Applying our results in Section 3.3, 1 A D C C D 2 H 5 : 2 C C 1 9 C Direct calculation shows that E 3 D A mod 9/: Applying Proposition 4.3, we have ŒH 5 W H.3 m / D m 1/ for m 1: (5.4)
11 Commutator subgroup and index formula of the Hecke group H N./ is a power of p, where 3 p takes the form 10k 3 It follows that p is a prime in ZŒ and D p n. Note first that ŒH 5 W H.p/ D jsl.2; p 2 /j: By Lemma 3.3 (m D p), 1 12p T 20p S 2 H.p/=H.p 2 /: By Proposition 4.3, ŒH 5 W H.p n / D p 6.n 1/ jsl.2; p 2 /j D N.p n / 3.1 N.p/ 2 /: (5.5) 5.5 N./ is a power of p, where p takes the form 10k 1 Observe that p is not a prime in ZŒ and p D, where and are primes in ZŒ. Consequently, D p r s. By Lemma 3.3 (m D p), 1 12p T 20p S 2 H.p/=H.p 2 /: By Proposition 4.3, It follows immediately from (2.2) that In summary, ŒH.p m / W H.5; p mc1 / D p 6 : ŒH. m p n / W H.5; mc1 p n / D p 3 for m 1: ŒH.p m / W H.p mc1 / D p 6 ; ŒH. m p n / W H. mc1 p n / D p 3 for m 1: (5.6) Lemma 5.1. Let p D 10k 1 2 N be a rational prime and let p D, where and are primes in ZŒ. Then ŒH 5 W H.p/ D ŒH 5 W H./ ŒH 5 W H./ D..p 1/p.p C 1// 2 : Proof. Let H 0.p/ be given as in (1.2). Applying results of [2] and [9], (i) ŒH 5 W H 0.p/ D.p C 1/ 2 (see [2, Lemma 1]), (ii) H./ \ H./ D H.p/, ŒH 5 W H./ D ŒH 5 W H./ D.p 1/p.p C 1/, (iii) G 5 =G./ Š G 5 =G./ Š PSL.2; p/ is simple (see [9, Corollary 2]). By (i) (ii) of the above, H./ H./. By (iii) of the above, H./H./ D H 5. Applying the Second Isomorphism Theorem, one has ŒH 5 W H.p/ D ŒH 5 W H./ ŒH 5 W H./ : Hence ŒH 5 W H.p/ D..p 1/p.p C 1// 2.
12 86 C. L. Lang and M. L. Lang The index ŒH 5 W H.p r s / can be determined by applying (5.6) and Lemma 5.1. In summary, ŒH 5 W H./ D N./ Y 3.1 N.P / 2 /: (5.7) 6 The main results P j./ Let X 2 ZŒ. Consider the decomposition X D Q i x i, where N.x i / is the power of a rational prime p i such that gcd.p i ; p j / D 1 for all i j. By Lemma 4.1, ŒH 5 W H.X/ D Q i ŒH 5 W H.x i /. The index ŒH 5 W H.x i / can be determined by applying (5.2), (5.3), (5.4), (5.5), and (5.7). In short, we have, if gcd.n./; 6/ D 1, then ŒH 5 W H.2 a 3 b / D I a J b N./ Y 3.1 N.P / 2 /; (6.1) P j./ where the product is over the set of all prime ideals that divide, I 0 D 1, I 1 D 10, I a D a 1/ if a 2, J 0 D 1, J b D b 1/ if b 1. As a corollary of formula (6.1), one has the following. Corollary 6.1. Let A be an ideal. Then the natural homomorphism f W H 5 SL.2; ZŒ =A/ is surjective if and only if the norm of A is prime to 6. 7 Application: The commutator subgroup of H 5 is not congruence Applying our results in Section 5.1, we have the following: H 5 =H.5/ Š H. C 2/=H.5/ H 5 =H. C 2/ Š E 5 3SL.2; 5/; where H. C 2/=H.5/ Š E 3 5 Š Z 5 Z 5 Z 5 is the elementary abelian group of order 5 3 and H 5 =H. C 2/ Š SL.2; 5/, H. C 2/=H.5/ Š ha; b; ci D h i, where 4 0 a I C. C 2/ ; b I C. C 2/ ; (7.1) c I C. C 2/ : 0 4
13 Commutator subgroup and index formula of the Hecke group H H 5 5 and H 0 are not congruence 5 Denote by H 5 5 the subgroup of H 5 generated by all the elements of the forms x 5, where x 2 H 5. Then H 5 5 is known as the power subgroup of H 5 (see [1]). It is clear that H 5 5 is normal and that S; T 5 2 H 5 5. Further, the index of H 5 5 in H 5 is 5. Since H 5 =H 5 5 is abelian, H 5 5 contains the commutator subgroup H 0 5. Lemma 7.1. Suppose that H 5 5 is congruence. Then H.5/ H 5 5. Proof. Suppose that H5 5 is congruence. Then H.5m r/ H5 5 for some 5m r 2 N, where m 1 and gcd.5; r/ D 1. Let K be the smallest normal subgroup that contains H.5 m r/ and T 5m. It is clear that K H.5 m / \ H5 5. Since gcd.5; r/ D 1, T 5m 2 K and T r 2 H.r/, it follows that T 2 KH.r/ H.5 m /H.r/: Note that KH.r/ is a normal subgroup of H 5 and that T 2 KH.r/. Applying the proof of Lemma 4.1, one has H 5 D KH.r/. Note that H.5 m /H.r/ D H 5 and that H.5 m / \ H.r/ D H.5 m r/. By the Second Isomorphism Theorem, K=H.5 m r/ H.r/=H.5 m r/ D KH.r/=H.5 m r/ D H 5 =H.5 m r/ D H.5 m /=H.5 m r/ H.r/=H.5 m r/: Hence K D H.5 m /. As a consequence, H.5 m / D K H5 5. Let m be the smallest positive integer such that H.5 m / H5 5. Suppose that m 2. Applying our main result (6.1), H 5 =H.5 m / Š SL.2; ZŒ =5 m / Š SL.2; ZŒ /=L.5 m / (see Section 2 for notation). Hence H 5 =H.5 m / and SL.2; ZŒ /=L.5 m / have the same coset representatives. Since 1 1 is a coset representative of the quotient SL.2; ZŒ /=L.5 m /, there exists some 2 H 5 such that 1 1.mod 5 m /: Since m 2, one has 5m 1 2 H5 5. As a consequence, H 5 5=H.5m / contains the elements T 5m 1 D 1 5m 1 and 1 5 m 1 5m 1 2 H.5 m 1 /=H.5 m / \ H5 5 =H.5m /:
14 88 C. L. Lang and M. L. Lang Now we consider the subgroup V of H5 5=H.5m / generated by T 5m 1 and 5m 1. Applying the proof of Proposition 4.3, we obtain that V D H.5 m 1 /=H.5 m /. In particular, we have H.5 m 1 / H5 5. This contradicts the minimality of m. Hence m D 1 and H.5/ H5 5. Proposition 7.2. The subgroups H5 5 and H 5 0 are not congruence. Proof. Since H5 0 H 5 5, it suffices to show that H 5 5 is not congruence. Suppose that H5 5 is congruence. By Lemma 7.1, H.5/ H 5 5. As H 5=H.C2/ Š SL.2; 5/ has no normal subgroup of index 5 and H5 5 has index 5 in H 5, it follows that H. C 2/ is not a subgroup of H5 5. This implies that H 5 5H. C 2/ D H 5. By the Second Isomorphism Theorem, jh. C 2/=ŒH 5 5 \ H. C 2/ j D 5 and jœh 5 5 \ H. C 2/ =H.5/j D 52 : Note that E 5 3SL.2; 5/ Š H 5 =H.5/ acts on D D ŒH 5 5 \ H. C 2/ =H.5/ Š Z 5 Z 5 by conjugation. Note also that D is a subgroup of h i (see (7.1)). Recall that J D 2 Aut H 5 : 1 0 The subgroup D is invariant under the conjugation of J and every element of H 5 (in particular, S and T ). However, one sees by direct calculation that the only nontrivial subgroup of h i invariant under J, S, and T is h i itself (see Appendix B). A contradiction. Hence H5 5 is not congruence. 7.2 Discussion Denote by M r the number of subgroups of G 5 D H 5 =Z 2 of index r. Applying the results of [7], we get M 2 D 1, M 3 D M 4 D 0 and M 5 D 26. The only subgroup V of index 2 is generated by ST and TS (see Section 1.1 for notations). Since G 5 =G.2/ Š D 10 has a subgroup of index 2, V contains G.2/. Hence V is congruence of level 2. It follows from Proposition 7.2 that G 5 5 D hx5 W x 2 G 5 i is non-congruence of index 5, level 5 (see [6]). Hence, the smallest index of a non-congruence subgroup of G 5 is 5. As a consequence, the smallest index of a non-congruence subgroup of H 5 is also 5. Note that the smallest index of a noncongruence subgroup of SL.2; Z/ is 7.
15 Commutator subgroup and index formula of the Hecke group H 5 89 A Appendix A Lemma A.1. Let p 2 N be a prime. Then modulo p nc1 generates a group of order p 6, where is given as ± D 1 p n i 1 0 ; p n i 1 ; 1 p n ic1 p n ic2 W i D 0; 1 : (A.1) p n i 1Cp n ic1 Proof. Put the matrices in (A.1) into the forms I C p n U and I C p n V. One sees easily that.i C p n U /.I C p n V / I C p n.u C V /.mod p nc1 /: (A.2) Hence modulo p nc1 generates an abelian group. Note that (A.2) makes the multiplication of I C p n U and I C p n V into the addition of U and V. In order to show generates a group of order p 6 modulo p nc1, we consider the groups * + M D X i D 1 pn i 1 0 ; Y i D p n i ; 1 N D * Z i D 1 pn ic1 p n ic2 p n i 1 C p n ic1 It is easy to see that M and N are abelian groups of order p 4 and p 2 respectively. Applying (A.2) and the fact that N is abelian, the elements in N take the following simple form: Z c 0 0 Zc P pn 1 id0 c i ic1 p n P 1 id0 c i ic2 p n P 1 id0 c i i 1 C p n P 1 id0 c i ic1 Note that we may assume that 0 c i p take the form X a 0 0 X a 1 1 Y b 0 0 Y b 1 1 : : (A.3) 1. Similar to (A.3), the elements in M 1 p n P 1 id0 a i i p n P 1 id0 b : (A.4) i i 1 Suppose that we have Z c 0 0 Zc 1 1 X a 0 0 X a 1 1 Y b 0 0 Y b 1 1.mod p nc1 /. An easy study of the.22/-entries of (A.3) and (A.4) implies that 1X 1 C p n c i ic1 1.mod p nc1 /: id0 Hence c 0 D c 1 D 0. As a consequence, M \ N D ¹1º. Hence jhij D jm jjn j D p 6 :
16 90 C. L. Lang and M. L. Lang B Appendix B Lemma B.1. Let D C 2 and let D ¹a; b; cº, where a; b; c are given as in (7.1). Then the only nontrivial subgroup of h i invariant under the action of S, T and J is h i. Proof. Since.I C U /.I C V / I C.U C V /.mod 5/; multiplication of.i CU /.I CV / can be transformed into addition of U and V. This makes the multiplication of matrices a, b, and c easy. Consequently, one has r D.ac/.ab/ I C 0 0 ; 3 0 s D.ac/.ab/ 1 I C 0 3 ; t D bc I C : 0 3 It is clear that h i D ha; b; ci D hr; s; ti: Let A; B 2 G 5. Set A B D BAB 1. Direct calculation shows that r S D s 1 ; r T D rs 1 t 2 ; r J D s; s S D r 1 ; s T D s; s J D r; t S D t 1 ; t T D st; t J D t 1 : (B.1) Denote by M a nontrivial subgroup of hr; s; ti that is invariant under the conjugation of J, S and T. Let 1 D r i s j t k 2 M. One sees easily that: (i) If k 6 0 (mod 5), without loss of generality, we may assume that k D 1. Then J S D t 2 2 M. It follows that t 2 M. Hence Consequently, s 2 M. This implies In summary, r; s; t 2 M. t T D st 2 M : s S D r 1 2 M :
17 Commutator subgroup and index formula of the Hecke group H 5 91 (ii) If k 0 (mod 5), then takes the form r i s j. Suppose that i 0 (mod 5). Then 1 s j 2 M. It follows that s 2 M. Consequently, Hence r D s T 2 M : rs 1 t 2 D r T 2 M : As a consequence, t 2 M. In summary, r; s; t 2 M. In the case i 6 0 (mod 5), we may assume that i D 1. Hence rs j 2 M. It follows that.rs j / T.rs j / 1 D s 1 t 2 2 M : Consequently,.s 1 t 2 / T D st 2 2 M. This implies that.s 1 t 2 /.st 2 / D t 4 2 M. Hence t 2 M. One now sees easily that r; s; t 2 M. Hence the only nontrivial subgroup of h i invariant under J, S and T is h i itself. Acknowledgments. We would like to thank the anonymous referee for pointing out a mistake in a previous version of the proof of Lemma 7.1. Bibliography [1] I. N. Cangul, R. Sahin, S. Ikikardes and O. Koruoglu, Power subgroups of some Hecke groups II, Houston J. Math. 33 (2007), no. 1, [2] S. P. Chan, M. L. Lang, C. H. Lim and S. P. Tan, The invariants of the congruence subgroups G 0.P / of the Hecke group, Illinois J. Math. 38 (1994), [3] P. M. Cohn, On the structure of GL 2 of a ring, Publ. Math. Inst. Hautes Études Sci. 30 (1966), [4] R. S. Kulkarni An arithmetic-geometric method in the study of subgroups of the modular group, Amer. J Math. 113 (1991), [5] C. L. Lang and M. L. Lang, Wohlfahrt s theorem and index formula for elementary matrix groups and SL.2; O/, preprint (2014), [6] C. L. Lang and M. L. Lang, Wohlfahrt s theorem for the Hecke group G 5, preprint (2014), [7] M. L. Lang, C. H. Lim and S. P. Tan, Subgroups of the Hecke groups with small index, Linear Multilinear Algebra 35 (1993), [8] M. L. Lang, C. H. Lim and S. P. Tan, Independent generators for congruence subgroups of Hecke groups, Math. Z. 220 (1995),
18 92 C. L. Lang and M. L. Lang [9] M. L. Lang, C. H. Lim and S. P. Tan, Principal congruence subgroups of the Hecke groups, J. Number Theory 85 (2000), [10] A. Leutbecher, Über die Heckeschen Gruppen G./, Abh. Math. Semin. Univ. Hambg. 31 (1967), [11] A. Leutbecher, Über die Heckeschen Gruppen G./, II, Math. Ann. 211 (1974), [12] A. Ogg, Modular Forms and Dirichlet Series, W. A. Benjamin, New York, [13] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J. 21 (1954), [14] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, Received May 19, 2014; revised September 16, Author information Cheng Lien Lang, Department of Mathematics, I-Shou University, Kaohsiung, Taiwan. cllang@isu.edu.tw Mong Lung Lang, Singapore , Republic of Singapore. lang2to46@gmail.com
Chapter 5. Modular arithmetic. 5.1 The modular ring
Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence
More informationMATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.
MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same
More informationMath 547, Exam 1 Information.
Math 547, Exam 1 Information. 2/10/10, LC 303B, 10:10-11:00. Exam 1 will be based on: Sections 5.1, 5.2, 5.3, 9.1; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationCOMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635
COMP239: Mathematics for Computer Science II Prof. Chadi Assi assi@ciise.concordia.ca EV7.635 The Euclidean Algorithm The Euclidean Algorithm Finding the GCD of two numbers using prime factorization is
More information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationMath 2070BC Term 2 Weeks 1 13 Lecture Notes
Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationMathematics for Cryptography
Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1
More informationON THE LIFTING OF HERMITIAN MODULAR. Notation
ON THE LIFTING OF HERMITIAN MODULAR FORMS TAMOTSU IEDA Notation Let be an imaginary quadratic field with discriminant D = D. We denote by O = O the ring of integers of. The non-trivial automorphism of
More informationD-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 3. Modular arithmetic, quotients, product groups
D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 3 Modular arithmetic, quotients, product groups 1. Show that the functions f = 1/x, g = (x 1)/x generate a group of functions, the law of composition
More informationDIHEDRAL GROUPS II KEITH CONRAD
DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an infinite
More informationThe Connections Between Continued Fraction Representations of Units and Certain Hecke Groups
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 33(2) (2010), 205 210 The Connections Between Continued Fraction Representations of Units
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationFIXED-POINT FREE ENDOMORPHISMS OF GROUPS RELATED TO FINITE FIELDS
FIXED-POINT FREE ENDOMORPHISMS OF GROUPS RELATED TO FINITE FIELDS LINDSAY N. CHILDS Abstract. Let G = F q β be the semidirect product of the additive group of the field of q = p n elements and the cyclic
More informationMATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM
MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is
More information1 Overview and revision
MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationarxiv:math/ v1 [math.nt] 21 Sep 2004
arxiv:math/0409377v1 [math.nt] 21 Sep 2004 ON THE GCD OF AN INFINITE NUMBER OF INTEGERS T. N. VENKATARAMANA Introduction In this paper, we consider the greatest common divisor (to be abbreviated gcd in
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationMath 547, Exam 2 Information.
Math 547, Exam 2 Information. 3/19/10, LC 303B, 10:10-11:00. Exam 2 will be based on: Homework and textbook sections covered by lectures 2/3-3/5. (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of
More information4 Powers of an Element; Cyclic Groups
4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)
More informationMATH 145 Algebra, Solutions to Assignment 4
MATH 145 Algebra, Solutions to Assignment 4 1: a) Find the inverse of 178 in Z 365. Solution: We find s and t so that 178s + 365t = 1, and then 178 1 = s. The Euclidean Algorithm gives 365 = 178 + 9 178
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationGroups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002
Background on Groups, Rings, and Finite Fields Andreas Klappenecker September 12, 2002 A thorough understanding of the Agrawal, Kayal, and Saxena primality test requires some tools from algebra and elementary
More informationRINGS: SUMMARY OF MATERIAL
RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered
More informationMA4H9 Modular Forms: Problem Sheet 2 Solutions
MA4H9 Modular Forms: Problem Sheet Solutions David Loeffler December 3, 010 This is the second of 3 problem sheets, each of which amounts to 5% of your final mark for the course This problem sheet will
More information2. THE EUCLIDEAN ALGORITHM More ring essentials
2. THE EUCLIDEAN ALGORITHM More ring essentials In this chapter: rings R commutative with 1. An element b R divides a R, or b is a divisor of a, or a is divisible by b, or a is a multiple of b, if there
More informationCHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and
CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)
More informationMATH 361: NUMBER THEORY FOURTH LECTURE
MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the
More informationTripotents: a class of strongly clean elements in rings
DOI: 0.2478/auom-208-0003 An. Şt. Univ. Ovidius Constanţa Vol. 26(),208, 69 80 Tripotents: a class of strongly clean elements in rings Grigore Călugăreanu Abstract Periodic elements in a ring generate
More informationEUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972)
Intro to Math Reasoning Grinshpan EUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972) We all know that every composite natural number is a product
More informationLecture 7.5: Euclidean domains and algebraic integers
Lecture 7.5: Euclidean domains and algebraic integers Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More information120A LECTURE OUTLINES
120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication
More informationPublic-key Cryptography: Theory and Practice
Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationSolutions of exercise sheet 8
D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra
More informationCONSEQUENCES OF THE SYLOW THEOREMS
CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.
More informationGalois theory (Part II)( ) Example Sheet 1
Galois theory (Part II)(2015 2016) Example Sheet 1 c.birkar@dpmms.cam.ac.uk (1) Find the minimal polynomial of 2 + 3 over Q. (2) Let K L be a finite field extension such that [L : K] is prime. Show that
More information3 The fundamentals: Algorithms, the integers, and matrices
3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers
More informationAlgebra SEP Solutions
Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationAlgebra for error control codes
Algebra for error control codes EE 387, Notes 5, Handout #7 EE 387 concentrates on block codes that are linear: Codewords components are linear combinations of message symbols. g 11 g 12 g 1n g 21 g 22
More informationLECTURE NOTES IN CRYPTOGRAPHY
1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics SOME PROPERTIES OF THE PROBABILISTIC ZETA FUNCTION OF FINITE SIMPLE GROUPS Erika Damian, Andrea Lucchini, and Fiorenza Morini Volume 215 No. 1 May 2004 PACIFIC JOURNAL OF
More informationNONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS CARRIE E. FINCH AND LENNY JONES Abstract. Let G be a finite group and let x G. Define the order subset of G determined by x to be the set of all elements in
More informationElementary Algebra Chinese Remainder Theorem Euclidean Algorithm
Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. This is presented in more depth than what we
More informationDefinition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson
Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.
More informationHamburger Beiträge zur Mathematik
Hamburger Beiträge zur Mathematik Nr. 270 / April 2007 Ernst Kleinert On the Restriction and Corestriction of Algebras over Number Fields On the Restriction and Corestriction of Algebras over Number Fields
More informationSlides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking
More informationOn the Frobenius Numbers of Symmetric Groups
Journal of Algebra 221, 551 561 1999 Article ID jabr.1999.7992, available online at http://www.idealibrary.com on On the Frobenius Numbers of Symmetric Groups Yugen Takegahara Muroran Institute of Technology,
More informationMATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology.
MATH 501 Discrete Mathematics Lecture 6: Number theory Prof. Dr. Slim Abdennadher, slim.abdennadher@guc.edu.eg German University Cairo, Department of Media Engineering and Technology 1 Number theory Number
More informationMath 451, 01, Exam #2 Answer Key
Math 451, 01, Exam #2 Answer Key 1. (25 points): If the statement is always true, circle True and prove it. If the statement is never true, circle False and prove that it can never be true. If the statement
More informationSchool of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information
MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon
More information(Rgs) Rings Math 683L (Summer 2003)
(Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that
More informationNOTES ON SIMPLE NUMBER THEORY
NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,
More informationMath 4400, Spring 08, Sample problems Final Exam.
Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that
More informationCS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II
CS 5319 Advanced Discrete Structure Lecture 9: Introduction to Number Theory II Divisibility Outline Greatest Common Divisor Fundamental Theorem of Arithmetic Modular Arithmetic Euler Phi Function RSA
More informationCover Page. The handle holds various files of this Leiden University dissertation
Cover Page The handle http://hdl.handle.net/1887/37019 holds various files of this Leiden University dissertation Author: Brau Avila, Julio Title: Galois representations of elliptic curves and abelian
More informationK. Ireland, M. Rosen A Classical Introduction to Modern Number Theory, Springer.
Chapter 1 Number Theory and Algebra 1.1 Introduction Most of the concepts of discrete mathematics belong to the areas of combinatorics, number theory and algebra. In Chapter?? we studied the first area.
More informationThe primitive root theorem
The primitive root theorem Mar Steinberger First recall that if R is a ring, then a R is a unit if there exists b R with ab = ba = 1. The collection of all units in R is denoted R and forms a group under
More informationUniversal localization at semiprime Goldie ideals
at semiprime Goldie ideals Northern Illinois University UNAM 23/05/2017 Background If R is a commutative Noetherian ring and P R is a prime ideal, then a ring R P and ring homomorphism λ : R R P can be
More informationCourse 2316 Sample Paper 1
Course 2316 Sample Paper 1 Timothy Murphy April 19, 2015 Attempt 5 questions. All carry the same mark. 1. State and prove the Fundamental Theorem of Arithmetic (for N). Prove that there are an infinity
More informationCongruent Number Problem and Elliptic curves
Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using
More informationChapter 14: Divisibility and factorization
Chapter 14: Divisibility and factorization Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter
More informationTC10 / 3. Finite fields S. Xambó
TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the
More informationφ(a + b) = φ(a) + φ(b) φ(a b) = φ(a) φ(b),
16. Ring Homomorphisms and Ideals efinition 16.1. Let φ: R S be a function between two rings. We say that φ is a ring homomorphism if for every a and b R, and in addition φ(1) = 1. φ(a + b) = φ(a) + φ(b)
More informationProf. Ila Varma HW 8 Solutions MATH 109. A B, h(i) := g(i n) if i > n. h : Z + f((i + 1)/2) if i is odd, g(i/2) if i is even.
1. Show that if A and B are countable, then A B is also countable. Hence, prove by contradiction, that if X is uncountable and a subset A is countable, then X A is uncountable. Solution: Suppose A and
More informationCongruences and Residue Class Rings
Congruences and Residue Class Rings (Chapter 2 of J. A. Buchmann, Introduction to Cryptography, 2nd Ed., 2004) Shoichi Hirose Faculty of Engineering, University of Fukui S. Hirose (U. Fukui) Congruences
More informationQuizzes for Math 401
Quizzes for Math 401 QUIZ 1. a) Let a,b be integers such that λa+µb = 1 for some inetegrs λ,µ. Prove that gcd(a,b) = 1. b) Use Euclid s algorithm to compute gcd(803, 154) and find integers λ,µ such that
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationIIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1
IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1 Let Σ be the set of all symmetries of the plane Π. 1. Give examples of s, t Σ such that st ts. 2. If s, t Σ agree on three non-collinear points, then
More informationHow many units can a commutative ring have?
How many units can a commutative ring have? Sunil K. Chebolu and Keir Locridge Abstract. László Fuchs posed the following problem in 960, which remains open: classify the abelian groups occurring as the
More informationComputations/Applications
Computations/Applications 1. Find the inverse of x + 1 in the ring F 5 [x]/(x 3 1). Solution: We use the Euclidean Algorithm: x 3 1 (x + 1)(x + 4x + 1) + 3 (x + 1) 3(x + ) + 0. Thus 3 (x 3 1) + (x + 1)(4x
More informationTHE GROUP OF UNITS OF SOME FINITE LOCAL RINGS I
J Korean Math Soc 46 (009), No, pp 95 311 THE GROUP OF UNITS OF SOME FINITE LOCAL RINGS I Sung Sik Woo Abstract The purpose of this paper is to identify the group of units of finite local rings of the
More informationALGEBRA HANDOUT 2.3: FACTORIZATION IN INTEGRAL DOMAINS. In this handout we wish to describe some aspects of the theory of factorization.
ALGEBRA HANDOUT 2.3: FACTORIZATION IN INTEGRAL DOMAINS PETE L. CLARK In this handout we wish to describe some aspects of the theory of factorization. The first goal is to state what it means for an arbitrary
More informationMath 312/ AMS 351 (Fall 17) Sample Questions for Final
Math 312/ AMS 351 (Fall 17) Sample Questions for Final 1. Solve the system of equations 2x 1 mod 3 x 2 mod 7 x 7 mod 8 First note that the inverse of 2 is 2 mod 3. Thus, the first equation becomes (multiply
More informationLecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman
Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Factorization 0.1.1 Factorization of Integers and Polynomials Now we are going
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. (a) Circle the prime
More informationPrincipal congruence subgroups of the Hecke groups and related results
Bull Braz Math Soc, New Series 40(4), 479-494 2009, Sociedade Brasileira de Matemática Principal congruence subgroups of the Hecke groups and related results Sebahattin Ikikardes, Recep Sahin and I. Naci
More informationIUPUI Qualifying Exam Abstract Algebra
IUPUI Qualifying Exam Abstract Algebra January 2017 Daniel Ramras (1) a) Prove that if G is a group of order 2 2 5 2 11, then G contains either a normal subgroup of order 11, or a normal subgroup of order
More informationSUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More information1. Factorization Divisibility in Z.
8 J. E. CREMONA 1.1. Divisibility in Z. 1. Factorization Definition 1.1.1. Let a, b Z. Then we say that a divides b and write a b if b = ac for some c Z: a b c Z : b = ac. Alternatively, we may say that
More information4 Number Theory and Cryptography
4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic This section introduces the basics of number theory number theory is the part of mathematics involving integers and their properties.
More informationMath.3336: Discrete Mathematics. Primes and Greatest Common Divisors
Math.3336: Discrete Mathematics Primes and Greatest Common Divisors Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationGroup Theory (Math 113), Summer 2014
Group Theory (Math 113), Summer 2014 George Melvin University of California, Berkeley (July 8, 2014 corrected version) Abstract These are notes for the first half of the upper division course Abstract
More informationCHAPTER 6. Prime Numbers. Definition and Fundamental Results
CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n
More informationIntroduction to finite fields
Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in
More informationMATH FINAL EXAM REVIEW HINTS
MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationECEN 5022 Cryptography
Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,
More informationLemma 1.1. The field K embeds as a subfield of Q(ζ D ).
Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms
More informationMath Introduction to Modern Algebra
Math 343 - Introduction to Modern Algebra Notes Field Theory Basics Let R be a ring. M is called a maximal ideal of R if M is a proper ideal of R and there is no proper ideal of R that properly contains
More informationModular Arithmetic and Elementary Algebra
18.310 lecture notes September 2, 2013 Modular Arithmetic and Elementary Algebra Lecturer: Michel Goemans These notes cover basic notions in algebra which will be needed for discussing several topics of
More information