Coset Diagrams Research Articles

Coset diagrams in the study of finitely presented groups with an application to quotients of the modular group

Coset diagrams in the study of finitely presented groups with an application to quotients of the modular group

A 5 as a Homomorphic Image of a Subgroup of Picard Group

In this article we investigate that what actions of a subgroup G 1 of Picard group yield A 5 = PSL(2, 5). We draw coset diagrams (propounded by G. Higman) for the action of G 1 on finite fields PL(F 5 n ). The projective special linear groups PSL(2, p) when p = 5 can be seen as 5-multiples of tetrahedral rotation point group. It has applications in carbon chemistry and physics as PSL(2, 5) is the rotation group of the fullerene C 60.

Cayley graphs, Cori hypermaps, and dessins d'enfants

This paper explains some facts probably known to experts and implicitely contained in the literature about dessins d'enfants but which seem to be nowhere explicitely stated. The 1-skeleton of every regular Cori hypermap is the Cayley graph of its automorphism group, embedded in the underlying orientable surface. Conversely, every Cayley graph of a finite two-generator group has an embedding as the 1-skeleton of a regular hypermap in the Cori representation. For non-regular hypermaps there is an analogous correspondence with Schreier coset diagrams.

The Number of Subgroups of PSL(2, Z) When Acting on F p ∪ {∞}

Coset diagrams defined for the transitive actions of PSL(2, Z) on projective line over a Galois field F p , PL(F p ), where p is prime, are used to obtain a formula for finding the number of subgroups of index p + 1 of the modular group PSL(2, Z). Some intransitive actions of PSL(2, Z) on PL(F p ) for some special values of θ, when θ ∈ F p , are also studied.

Coset Diagrams for a Homomorphic Image of Δ (3, 3, k)

Let q be a prime power. By PL(F q) the authors mean a projective line over the finite field F q with the additional point ∞. In this article, the authors parametrize the conjugacy classes of nondegenerate homomorphisms which represent actions of Δ (3, 3, k) =〈 u, v:u 3= v 3=( uv) k =1〉 on PL( F q ), where q≡ ± 1(mod k). Also, for various values of k, they find the conditions for the existence of coset diagrams depicting the permutation actions of Δ (3, 3, k) on PL( F q ). The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by ( u ¯ v ¯ ) k . In this way, they get Δ (3, 3, k) as permutation groups on PL( F q ).

Inner reflectors and non-orientable regular maps

Regular maps on non-orientable surfaces are considered with particular reference to the properties of inner reflectors, corresponding to symmetries of the 2-fold smooth orientable covering which project onto local reflections of the map itself. An example is given where no inner reflector is induced by an involution, and the existence of such involutions is related to questions of symmetry of coset diagrams for the symmetry group of the map.

Homomorphic images of [formula omitted

It is known that each conjugacy class of actions of PGL ( 2 , Z ) on F q ∪ { ∞ } can be represented by a coset diagram D ( θ , q ) , where θ ∈ F q and q is a power of a prime p. In this paper, we are interested in parametrizing the conjugacy classes of actions of the infinite triangle group △ ( 2 , 3 , 11 ) = 〈 x , y : x 2 = y 3 = ( xy ) 11 = 1 〉 on F q ∪ { ∞ } . For each θ ∈ F q we then associate a coset diagram D ( θ , q ) depicting the conjugacy class of actions of △ ( 2 , 3 , 11 ) on F q ∪ { ∞ } . We have obtained conditions on θ and q which guarantee only those coset diagrams which depict homomorphic images of △ ( 2 , 3 , 11 ) in PGL ( 2 , q ) . We are interested in finding also when the coset diagrams for the actions of PGL ( 2 , Z ) on F q ∪ { ∞ } contain vertices on the vertical line of symmetry. It will enable us to show that for infinitely many values of q , the group PGL ( 2 , q ) has minimal genus, while also for infinitely many q , the group PSL ( 2 , q ) is an H * -group.

Group generated by two elements of orders two and six acting on [formula omitted] and [formula omitted

There is a well-known relation between the action z → ( az + b)/( cz + d) of the modular group on the real line R and continued fractions. In this paper we replace the modular group by M = 〈 x, y : x 2 = y 6 = 1〉 and R by rational projective line and real quadratic field. We define coset diagrams for the orbits of M acting on these fields and show that in the orbit pM, where p = ( a + √ n)/ c, the non-square positive integer n does not change its value and the numbers of the form p, where p and its algebraic conjugate p = (a − √n)/c have different signs, are finite in number and that part of the coset diagram containing such numbers forms a single closed path and it is the only closed path in the orbit pM.

On word structure of the modular group over finite and real quadratic fields

Let Ω denote the projective line over the real quadratic field and δ denote the projective line over the finite field F q with q elements. Coset diagrams for the orbits of the modular group G acting on Ω and δ give some interesting information. By using these diagrams we determine a condition for the existence of an orbit of G on Ω containing a circuit of a given type. If such a circuit exists, we find a condition under which the orbit contains a real quadratic irrational number α along with its algebraic conjugate α ̄ . As there are two projections from Ω to δ we are interested in the case when G acts on δ and we determine necessary and sufficient conditions for the existence of two orbits of G: one containing α along with 1/α and the other containing α together with 1/ α ̄ .

A family of conformally asymmetric Riemann surfaces

AbstractWe give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g ≥ 3. The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3,r) for certain values of r.

Factor groups of G 6,6,6 through coset diagrams for an action on PL ( F q)

Let G=〈 x, y, t: x 2= y k = t 2=( xt) 2=( yt 2=1〉 and q be a prime power. Then any homomorphism from G into PGL(2, q) induces an action on the projective line over F q . Such an action can be depicted by a coset diagram. We show how the existence of certain types of fragments in these coset diagrams may be related to properties of a corresponding parameter ϑ= r 2/ βdG, where r and βdG are the trace and determinant of a matrix representing the image of xy in PGL(2, q). We also show how these fragments can be used to show that for a family of positive integers n, all A n and S n are quotients of G 6,6,6 .

Group generated by two elements of orders 2 and 4 acting on real quadratic fields

Coset diagrams for the orbit of the groupG=〈x,y∶x2=y4=1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that in the orbitpG, where\({{p = (a + \sqrt n )} \mathord{\left/ {\vphantom {{p = (a + \sqrt n )} c}} \right. \kern-\nulldelimiterspace} c}\), the non-square positive integern does not change its value and the real quadratic irrational numbers of the formp, wherep and its algebraic conjugate\({{(a - \sqrt n )} \mathord{\left/ {\vphantom {{(a - \sqrt n )} c}} \right. \kern-\nulldelimiterspace} c}\) have different signs, are finite in number and that part of the coset diagram containing such numbers forms a single closed path, which is the only closed path in the orbit ofp.

Generating the Mathieu groups and associated Steiner systems

With the aid of two coset diagrams which are easy to remember, it is shown how pairs of generators may be obtained for each of the Mathieu groups M 11, M 12, M 22, M 23 and M 24, and also how t is then possible to use these generators to construct blocks of the associated Steiner systems S(4, 5, 11), S(5, 6, 12), S(3, 6, 22), and S(5, 8, 24) respectively.

A question by Graham Higman concerning quotients of the (2, 3, 7) triangle group

Using coset diagrams it is shown that for every sufficiently large positive integer n, both the alternating group A n and the symmetric group S n are quotients of the group G 3,7, 168 = 〈A, B, C ¦ A 3 = B 7 = C 168 = (AB) 2 = (BC) 2 = (CA) 2 = (ABC) 2 = 1〉 , and hence that all but finitely many of the alternating groups A n can be generated by elements x and y satisfying the relations x 2 = y 3 = ( xy) 7 = ( x −1 y −1 xy) m = 1, where m = 84. This is the smallest known value of m for which this phenomenon occurs.

Coset diagrams for hurwitz groups

Hurwitz groups are non-trivial quotients of Various families of groups have been thoroughly investigated as candidates for Hurwitz groups. In this paper we are interested in parametrizing the actions of ▵(2,3,7) on the projective lines over finite fields Fq, with the help of coset diagrams. We are interested to know also the values of q for which there is a natural homomorphism induced from ▵(2,3,7) and there exist vertices on the vertical line of symmetry in the diagram depicting these actions. Also we use the Cebotarev's Density Theorem and Galois Theory to answer the question about the frequency with which such situations and certain fragments of the diagram occur.

Modular group acting on real quadratic fields

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.

On a result of Petersson concerning the modular group

SynopsisLet N(u, g, h) denote the number of subgroups of index u, genus g and parabolic class number h in the classical modular group. In 1974, Petersson proved that, for g = 0 and h = 1, 2, N(u, g, h) grows exponentially with u. We obtain a formula valid for all g and h. From this we derive an asymptotic estimate, showing that Petersson's result gives the correct type of growth. The proofs use coset diagrams.

Subgroups of finite index in a free product with amalgamated subgroup

Let G be a free product of finitely many finite groups with amalgamated subgroup. Using coset diagrams, a recurrence relation is obtained for the number of subgroups, and of free subgroups, of each finite index in G. In the latter case, an asymptotic formula is derived. When the amalgamated subgroup is central, the relation takes a simpler form.

Subgroups of the (2,3,7) triangle group

With a subgroup of finite index in the (2,3,7) triangle group, we associate a quintuple of non-negative integers (u,p,e,f,g), with u ≥1 and u=84(p−1)+21e+28f+36g. We show that, with three exceptions, each quintuple satisfying the conditions corresponds to a subgroup. The proof uses coset diagrams.

Impossible specifications for the modular group

With any subgroup of the modular group we associate a set of non-negative integers satisfying two conditions. This set is the specification of the subgroup. Several sets were known which satisfied both conditions, but which did not correspond to subgroups. Here we find several infinite families of such sets. We also investigate situations where a given set corresponds to just one conjugacy class of subgroups. These are related to the well-known lattice subgroups. The method involves the use of coset diagrams, described by Atkin and Swinnerton-Dyer in [1].