Finite permutation representation of a subgroup of picard group

Acta Mathematica Scientia 2012,32B(3):842–850 http://actams.wipm.ac.cn

FINITE PERMUTATION REPRESENTATION OF A SUBGROUP OF PICARD GROUP∗ Qaiser Mushtaq

Shahla Asif

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan E-mail: [email protected]; [email protected]

Abstract We investigate action of a subgroup G1 of the Picard group on finite sets using coset diagrams. We show that its actions on the sets of 3, 4, 5, 6, 8, and 12 elements yield building blocks of Coset diagrams and that these blocks can be connected together so that a diagram of n vertices can be obtained. We show that various combinations of these blocks represent alternating and symmetric groups of various degrees. We show also that the action of G1 on a set of n vertices is transitive. Key words Bianchi groups; Picard group; coset diagrams 2000 MR Subject Classification

1

20C30; 20B35; 05C20

Bianchi Groups

√ Let Od be the ring of algebraic integers of an imaginary quadratic field Q −d. A Bianchi group is the group Γd = P SL2 (Od ), where d is a positive square free integer. L. Bianchi studied these groups as a natural class of discrete subgroups of P SL2 (C). Picard studied the group Γ1 that is P SL2 (O1 ) = P SL2 (Z[i]), where O1 denotes the ring of Gaussian integers √ {a + bi = a + −1b : a, b ∈ Z}. Due to a method developed by P. M. Cohn [1], we can show that Γ1 has presentation 2

3

3

Γ1 = a, l, t, u; a2 = l2 = (al)2 = (tl)2 = (ul) = (at) = (ual) = [t, u] = 1, where a : z −→ −1 z , t : z −→ z + 1, u : z −→ z + i, l : z −→ −z, and the matrices related to these linear fractional transformations are, respectively, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −1 11 1i i 0    ⎠,T = ⎝ ⎠,U = ⎝ ⎠ , and L = ⎝ ⎠. A =⎝ 1 0 01 01 0 −i Using the Tietze transformations, namely, R = lau−1 , S = al, U −1 = ta, and V = a, we can obtain another presentation for Γ1 : ∗ Received

December 28, 2007.

No.3

Qaiser Mushtaq & Shahla Asif: FINITE PERMUTATION REPRESENTATION 2

2

2

843

2

R, S, U, V : R3 = S 2 = U 3 = V 2 = (RU ) = (RV ) = (SU ) = (SV ) = 1, where   1 R : z −→ z−i , and V : z −→ −1 , S : z −→ z1 , U : z −→ − 1+z z z . It is worth mentioning that 3 2 U, V : U = V = 1 is the modular group P SL(2, Z) which we denote by M . 2 2 Let G1 = R, U, V : R3 = U 3 = V 2 = (RU ) = (RV ) = 1 and G2 = S, U, V : S 2 = 2 2 U 3 = V 2 = (SU ) = (SV ) = 1. Then, G2 is the extended modular group P GL(2, Z). It is well known that Γ1 is a free product of its factor subgroups G1 and G2 with M amalgamated, that is, Γ1 ∼ = G1 ∗M G2 . 2 In this article, the object of our investigations is G1 = R, U, V : R3 = U 3 = V 2 = (RU ) = 2 (RV ) = 1. Our main purpose is to draw coset diagrams (propounded by Graham Higman) for the natural action of G1 on finite sets and study the action through them.

2

Action of G1 on Finite Sets

In [4], for the action of G1 on P L(Fp ), the projective line over the finite fields Fp , where p is the Pythagorean prime, we have used linear fractional transformations and obtained permutations to draw diagrams. Whereas in the case of finite sets, we do not need linear fractional transformations. We draw diagrams satisfying the relations of the group G1 . After having drawn the diagrams, we may label the vertices and form permutations. We observe that for the action of G1 on finite sets, the corresponding diagrams naturally have symmetry about the vertical line of axis. As we are not making use of linear fractional transformations so we denote this symmetry by the notation T which we could not find in the case of P L(Fp ) ([4] theorem 2). That is, an involution which satisfies the following relations T 2 = (RT )2 = (U T )2 = (V T )2 = 1. From now on, by G∗1 we mean R, U, V, T : R3 = U 3 = V 2 = T 2 = (RU )2 = (RV )2 = (RT )2 = (U T )2 = (V T )2 = 1. We construct coset diagrams for the action of the group G1 on finite sets. A coset diagram [3] for G∗1 is defined as follows. The 3-cycles of R and U are, respectively, denoted by the three sides of directed unbroken and doted edges forming triangles. Any two vertices which are interchanged by the involution V are denoted by an edge (not necessarily a straight line). Fixed points of R and U , if they exist, are denoted by loops and those of V by heavy dots. Whereas the involution T is depicted by the symmetry about the vertical line of axis.

844

ACTA MATHEMATICA SCIENTIA

Vol.32 Ser.B

For instance, the preceding diagram depicts a (transitive) permutation representation of G∗1 of degree 15, in which R acts as (a b c)(d k g)(e i j)(f p n)(h m l), U acts as (a c b)(d f e)(h k j)(n m i)(g l p), V acts as (a n)(b p)(c f )(d k)(e j)(l m)(i)(g)(h) while the permutation of T is (a b)(e d)(g i)(k j)(l m)(p n)(c)(f )(h). We require a method of joining smaller diagrams together so that a larger diagram of the required size can be obtained. Any two or more coset diagrams can be joined together to obtain a coset diagram of an arbitrary size provided they are stitched together in a special way. The new coset diagram thus produced will still preserve all the relations of G∗1 . There are six of these basic diagrams that are needed for this purpose. To connect a number of copies of these diagrams, one needs the following fragment of a coset diagram.

We call this fragment a ‘connector’ and denote it by Cca,b . So, by a connector Cca,b , we mean a fragment of a coset diagram containing vertices a, b, c, such that V (a) = b, V (c) = c, and (a, b, c) occurs in the permutation R. These fragments are present in all the diagrams except the diagram containing 8 elements. The diagram with 8 elements contains a different type of fragment, which we would discuss later. The method for stitching together the coset diagrams is explained as follows.   Consider two diagrams for G∗1 , each containing at least one connector Cca,b and Cca ,b . Place the two diagrams on a common vertical axis of symmetry, one above the other and add three V -edges. One joins a to b , the second b to a and the third c to c , as follows.

If one diagram has m vertices and the second has n vertices, then, the resulting diagram depicts a transitive permutation group of degree m + n generated by the elements R, U, V, T 2 2 satisfying the relations R3 = U 3 = V 2 = T 2 = (RU ) = (RV ) = (RT )2 = (U T )2 = (V T )2 = 1. In this way, we can construct the diagram for any number of vertices.

3

Basic Diagrams

We denote a coset diagram with n vertices by D(n). As we have mentioned earlier, the proof of our main result requires a list of just six diagrams, namely, D(3), D(4), D(5), D(6),

No.3

Qaiser Mushtaq & Shahla Asif: FINITE PERMUTATION REPRESENTATION

845

D(8), and D(12). We call these basic diagrams as ‘building blocks’. The diagram D(8) is a special one and a number of copies of D(8) are connected with themselves only. We shall explain D(8) as a special case later. For each building block, we write down a specification, which describes the extra relations satisfied by the diagram, the order of the group evolved and some additional information about the permutation group. We classify these building blocks as follows. Cl(1) This class contains three diagrams, one for each set of three, four and five elements, respectively. Each member of this class contains only one connector. So that we can join any diagram only once within this class.

D(3) represents a non-abelian and non-simple group of order 3! with defining relations 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )2 = (U RV )2 = (V RU )2 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )2 = (V U V )3 = 1. Letting x = R, y = V , the above relations reduce to x2 = y 3 = (xy)2 = 1. This shows that the diagram actually represents S3 , the symmetric group of degree 3.

D(4) represents a non-abelian and non-simple group of order 4! having the following: 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )2 = (U RV )4 = (V RU )4 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )2 = (V U V )3 = 1, as its defining relations. Letting x = R, y = U RV , we obtain a new presentation of the group, that is, x3 = y 4 = (xy)2 = 1 ∼ = S4 , the symmetric group of degree 4.

D(5) represents a non-abelian and simple group of order 5! having 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )3 = (U RV )5 = (V RU )5 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )3 = (V U V )3 = 1,

846

ACTA MATHEMATICA SCIENTIA

Vol.32 Ser.B

as its possible relations. Letting x = RV , y = U , the above relations yield the group x2 = y 3 = (xy)5 = 1 ∼ = A5 , the alternating group of degree 5. Cl(2) This class contains the following three diagrams for D(6) :

For the set of 6 elements, there are two number of connectors so that we can join at a time: (i) any two diagrams of cl(1), or (ii) any number of diagrams of cl(2). These diagrams represent a non-abelian and non-simple subgroup of A6 having 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )2 = (U RV )4 = (V RU )4 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )2 = (V U V )3 = 1, as its possible relations. Letting x = R, y = U RV , we get x3 = y 4 = (xy)2 = 1 ∼ = S4 of order 4!. This class contains also the following three diagrams for D(12). As D(12) contains four connectors, we can join at a time; (i) any four diagrams of cl(1), or (ii) any number of diagrams of cl(2).

This diagram depicts a non-abelian and non-simple subgroup of S12 having defining relations 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )2 = (U RV )4 = (V RU )4 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )2 = (V U V )3 = 1. If we suppose x = R, y = U RV , then the relations in U, V , and R reduce to x3 = y 4 = (xy)2 = 1, which yield S4 of order 4!

No.3

Qaiser Mushtaq & Shahla Asif: FINITE PERMUTATION REPRESENTATION

847

The following two diagrams depict a non-abelian and non-simple subgroup of A12 of order 96 having 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )4 = (U RV )4 = (V RU )4 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )4 = (V U V )3 = 1, as its possible relations.

Cl(3) It consists of a single diagram, namely, D(8).

D(8) represents a non-abelian and non-simple subgroup of A8 of order 4! having 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (U V )2 = (U RV )4 = (V RU )4 = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (U V U )2 = (V U V )3 = 1, ∼ S4 . as its possible relations. Letting x = R, y = U RV , we obtain x3 = y 4 = (xy)2 = 1 = We join any number of copies of D(8) with itself only. It has a different type of connector that we call a ‘bridge’. By Bdc we mean a bridge in D(8) containing two points c and d such that R(c) = c, R(d) = d, and V (c) = d and appear as distinct vertices of two different triangles of U . For any k ∈ N, we take k copies of D(8) having bridges Bdcm , where m = 0, 1, 2, · · · , k − 1, m respectively. We place these copies on the common horizontal axis of symmetry side by side. Then, we add V -edges (cm , dm+1 ) k in number where m+1 ≡ 0 (modk) and m = 0, 1, 2, · · · , k−1. In the following remarks, we observe some properties of these building blocks, and the diagrams obtained by joining them together. Remark 1 All the building blocks for D(12) have symmetries about vertical and horizontal axes.

848

ACTA MATHEMATICA SCIENTIA

Remark 2

Vol.32 Ser.B

If we join k copies of D(6) with each other, for k ∈ N, then,

(i) (U V )2k = (U RV )4k = 1, if k is odd, yielding non-abelian and non-simple subgroups of A6k . (ii) (U V )2k = (U RV )2k = 1, if k is even, yielding non-abelian and non-simple subgroups of S6k . Remark 3 4k

(i) (U V ) of A12k , and

If we connect k copies of D(12) with each other, then, = (U RV )4k = 1, if k is odd, yielding non-abelian and non-simple subgroups

(ii) (U V )4k = (U RV )4k = 1, if k is even, yield non-abelian and non-simple subgroups of S12k . Remark 4 The k copies of D(8), where, k ∈ N, evolves (U V )2k = (U RV )4k = 1, yielding non-abelian and non-simple subgroups of A8k . All these groups have order 24k 3 .

4

Main Results Theorem 1

G∗1 acts transitively on n ≥ 3 elements.

Proof For proof, it is convenient to discuss three cases, separately. Case 1

3 ≤ n ≤ 12.

Since we already have diagrams for n = 3, 4, 5, 6, 8, and 12, so, to obtain D(7), we stitch together D(3) and D(4). The coset diagram D(9) can be obtained by stitching together either D(3) with D(6) or D(4) with D(5). Similarly, D(10) can be constructed by stitching together D(4) and D(6) or simply by 2 copies of D(5). In exactly the same way, D(11) can be obtained by stitching together D(5) and D(6). Case 2

12 < n < 24.

Then, (i) For n = 13, stitch together D(3) and D(4) with D(6). (ii) For n = 14, stitch together either 2 copies of D(4) with D(6) or D(3) and D(5) with D(6). (iii) When n = 15, 16, and 17, stitch together D(3), D(4), D(5) separately with D(12) to get D(15), D(16), D(17), respectively. To construct D(18), either stitch together 3 copies of D(6) with each other or 2 copies of D(3) or D(6) with D(12). (iv) For n = 19, 20, · · · , 23. Stitch together D(3) and D(4) with D(12) to construct D(19). Similarly, we stitch together either, 2 copies of D(4) or D(3) and D(5) with D(12) for constructing D(20). For D(21), stitch together either D(4) and D(5) or D(3) and D(6) or 3 copies of D(3) with D(12). D(22) can be constructed by stitching together either 2 copies of D(5) or D(4) and D(6) or 2 copies of D(3) and D(4) with D(12). Stitch together D(5) and D(6) or 2 copies of D(4) and D(3) or D(5) and 2 copies of D(3) with D(12) for obtaining D(23). Case 3

n ≥ 24.

No.3

Qaiser Mushtaq & Shahla Asif: FINITE PERMUTATION REPRESENTATION

849

Suppose n = 12q + r, where q and r are whole numbers such that q ≥ 2 and 0 ≤ r ≤ 11. Here again there are three cases to discuss. (i) When r = 0, then join q copies of D(12). (ii) If r = 1 or 2, then take n = 12(q − 1) + (12 + r), where either 12 + r = 13 or 14. If 12 + r = 13, we stitch together D(3), D(4), D(6), or 2 copies of D(5) and D(3), or 3 copies of D(3) and D(4) or D(5) and 2 copies of D(4) with (q − 1) copies of D(12). When 12 + r = 14, we consider D(3), D(5), D(6) or 2 copies of D(4) and D(6) or D(4) and 2 copies of D(5) or 2 copies of D(4) and 2 copies of D(3) or D(5) and 3 copies of D(3) and stitch them together with (q − 1) copies of D(12). (iii) As in case 1, for 3 ≤ r ≤ 11, stitch together those diagrams which are constituent of D(r), with q copies of D(12). As an illustration, let us obtain a coset diagram for the action of G∗1 on the set of n = 74 elements. Example 1 Let n = 74. As n = 12 × 6 + 2, here r = 2, so following case 3, part (ii) of theorem 1, n = 12 × 5 + (12 + 2) = 12 × 5 + 14. That is, r = 14 and q = 5. Now, we stitch together either D(3), D(5), D(6) or 2 copies of D(4) and D(6) or D(4) and 2 copies of D(5) or 2 copies of D(4) and 2 copies of D(3) or D(5) and 3 copies of D(3) with 5 copies of D(12). Suppose 14 = 3 + 5 + 6. As D(3), D(5), and D(6) have, respectively, 1, 1 and 2 connectors, so if we join D(3) and D(5) with 2 connectors of D(6), then we will be left with no connector to join this with 5 copies of D(12). Therefore, we have to stitch together either D(3) (or D(5)) separately with upper most (lower most) connector of 5 copies of D(12), D(5) (or D(3)) with one connector of D(6). Then, these two diagrams D(63) (or D(65)) and D(11) (or D(9)) are stitch together with each other by the remaining connector of D(6) and one of the lower most (upper most) connector of 5-copies of D(12) to maintain the vertical symmetry. Corollary 1 Coset diagram for the action of G1 on a finite set of n ≥ 3 elements is connected. Proof Since all the basic diagrams are transitive with at least one connector, which helps them to connect with other diagrams. Therefore, the larger diagram thus obtained, by stitching together the connected basic diagrams, is also connected. Hence the action of G1 on any finite set of n-elements, for n ≥ 3 is transitive. Theorem 2 For the action of G1 on a finite set of n ≥ 3 elements, the permutation group thus obtained satisfies the relations: 2

2

R3 = U 3 = V 2 = (RU ) = (RV ) = (RU R)3 = (U RU )3 = (RV R)2 = (V RV )3 = (V U V )3 = 1, as a subgroup of Sn . 2 2 Proof As R3 = U 3 = V 2 = (RU ) = (RV ) = 1 are the relations of G1 , all these relations will be satisfied by the diagram. For the remaining relations, we consider (V U V )3 = V U (V V )U (V V )U V = V (U IU IU )V = V (U U U )V = V 2 = 1. Furthermore, since (RU )2 = I, so RU R = U −1 . But ord(U −1 ) = 3 implies that (RU R)3 = I. Similarly, (U RU )3 = (V RV )3 = I. Also, (RV )2 = I implies that RV R = V −1 , and because of V 2 = I, (RV R)2 = I.

850

ACTA MATHEMATICA SCIENTIA

Vol.32 Ser.B

Theorem 3 In the permutation representation of G1 on n-elements, where n ≥ 3, there exist h, j ∈ N such that (U RV )h = (U V )j = 1. Proof The action of G1 on a finite set of n elements, where n ≥ 3, yields a finite group, namely, a subgroup of the permutation group Sn . Evidently, U RV and U V are the elements of the group obtained. Therefore, the order of them must be finite too. Proposition 1 In the permutation representation of G1 on n ≥ 3 elements, (i) ord(U V ) = ord(U V U ), (ii) ord(U RV ) = ord(RV U ) = ord(V U R), (iii) ord(V RU ) = ord(RU V ) = ord(U V R), and (iv) ord(U V ) = ord(V U ). Proof (i) Suppose that, for any positive integer ko , ord(U V U ) = ko . Then, (U V U )ko = (U V U )(U V U )(U V U ) · · · (U V U ) = 1 (ko -terms) implies that U [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )](V U ) = 1((ko − 1)-terms of V U 2 ). ko −1

(V U 2 )ko −1 V U = U −1 . So that (V U 2 ) ko −1

(V U 2 )

V = U implies

V U 2 = 1, because U −1 = U 2 .

Therefore, ord(V U 2 ) = ko . Next, [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )](V U 2 ) = 1 (ko -terms) evolves [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )]V = U ((ko − 1)-terms of V U 2 ). [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )] = U V ((ko − 1)-terms of V U 2 ). [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )](V U 2 ) = V U ((ko − 1)-terms of V U 2 ). [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )]V = U V U ((ko − 2)-terms of V U 2 ). [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )] = U V U V ((ko − 2)-terms of V U 2 ). [(V U 2 )(V U 2 )(V U 2 ) · · · (V U 2 )] = (U V )2 ((ko − 2)-terms of V U 2 ). ko −1

(1-term of V U 2 ). Continuing in the same fashion, we ultimately have V U 2 = (U V ) ko −1 ko −1 So, V = (U V ) U. Thus, 1 = (U V ) U V , that is, ord(U V ) = ko . Hence, ord(U V U ) = ord(U V ). Proofs of (ii), (iii), and (iv) are similar. References [1] Cohn P M. A Presentation of SL2 for Euclidean quadratic imaginary number fields. Mathematika, 1968, 15: 156–163 [2] Fine B. Algebraic Theory of the Bianchi Groups. New York: Marcel Dekker Inc, 1989 [3] Mushtaq Q, Servatius H. Permutation representation of the symmetry groups of regular hyperbolic tesselatoins. Jour London Math Soc, 1993, 48(2): 77–86 [4] Mushtaq Q, Asif S. A5 as a homomorphic image of a subgroup of picard group. Comm Algebra, 2010, 38(10): 3897–3912 [5] Stothers W W. Subgroups of (2, 3, 7) triangle group. Manuscripta Math, 1977, 20: 323–334