Linear Fractional Group Research Articles

Enumeration of regular maps of given type on twisted linear fractional groups

AbstractWe enumerate orientably regular maps of any given type with automorphism group isomorphic to a twisted linear fractional group; these groups form the ‘other’ family in Zassenhaus' classification of finite sharply 3‐transitive groups.

Linear fractional group as Galois group

We compute all signatures of [Formula: see text] and [Formula: see text] which classify all orientation preserving actions of the groups [Formula: see text] and [Formula: see text] on compact, connected, orientable surfaces with orbifold genus [Formula: see text]. This classification is well-grounded in the other branches of Mathematics like topology, smooth and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.

Self-dual, self-Petrie-dual and Möbius regular maps on linear fractional groups

Regular maps on linear fractional groups PSL(2, q) and PGL(2, q) have been studied for many years and the theory is well-developed, including generating sets for the associated groups. This paper studies the properties of self-duality, self-Petrie-duality and Mobius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5, 5). The final section includes an enumeration of the PSL(2, q) maps for q ≤ 81 and a list of all the PSL(2, q) maps which have any of these special properties for q ≤ 49.

On finite simple groups acting on homology spheres with small fixed point sets

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets (“pseudofree action”) is the alternating group \(\mathbb {A}_5\) acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most \(d\)-dimensional fixed points sets, for a fixed \(d\). We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group \(\mathbb {A}_5\) in dimensions 2, 3 and 5, the linear fractional group \(\mathrm{PSL}_2(7)\) in dimension 5, and possibly the unitary group \(\mathrm{PSU}_3(3)\) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.

NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)

AbstractWe present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

Nonorientable regular maps over linear fractional groups

It is well known that for any given hyperbolic pair ( k , m ) there exist infinitely many regular maps of valence k and face length m on an orientable surface, with automorphism group isomorphic to a linear fractional group. A nonorientable analogue of this result was known to be true for all pairs ( k , m ) as above with at least one even entry. In this paper we establish the existence of such regular maps on nonorientable surfaces for all hyperbolic pairs.

On minimal actions of finite simple groups on homology spheres and Euclidean spaces

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2, p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.

Non-orientable Regular Maps of a Given Type over Linear Fractional Groups

It is known that for any given k and m such that 1/k + 1/m < 1/2 there exist infinitely many regular maps M of valence k and face length m on orientable surfaces such that the automorphism group of M is isomorphic to a linear fractional group over a finite field. We examine the pairs (k, m) for which this result can be extended to regular maps on non-orientable surfaces.

On finite simple and nonsolvable groups acting on closed 4–manifolds

We show that the only finite nonabelian simple groups to admit a locally linear, homologically trivial action on a closed simply connected 4-manifold M (or on a 4-manifold with trivial first homology) are the alternating groups A 5 , A 6 and the linear fractional group PSL(2, 7). (We note that for homologically nontrivial actions all finite groups occur.) The situation depends strongly on the second Betti number b 2 (M) of M and was known before if b 2 (M) is different from two, so the main new result concerns the case b 2 (M) = 2. We prove that the only simple group that occurs in this case is A 5 , and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

On finite simple groups acting on integer and mod 2 homology 3-spheres

We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z 2 -homology 3-sphere are the linear fractional groups PSL ( 2 , q ) , for an odd prime power q (and the dodecahedral group A 5 ≅ PSL ( 2 , 5 ) in the case of an integer homology 3-sphere), by showing that G has dihedral Sylow 2-subgroups and applying the Gorenstein–Walter classification of such groups. We also discuss the minimal dimension of a homology sphere on which a linear fractional group PSL ( 2 , q ) acts.

On finite simple and nonsolvable groups acting on homology 4-spheres

The only finite non-Abelian simple group acting on a homology 3-sphere—necessarily non-freely—is the dodecahedral group A 5 ≅ PSL ( 2 , 5 ) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A 5 ∗ ≅ SL ( 2 , 5 ) ). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups A 5 ≅ PSL ( 2 , 5 ) and A 6 ≅ PSL ( 2 , 9 ) . From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-sphere.

Finite Presentation of a Linear-Fractional Group

In this paper, we show that the coset diagram for the action of a linear-fractional group generated by [Formula: see text] and [Formula: see text] on the rational projective line is transitive and the coset diagram for the action is connected. Using the coset diagram, we show that [Formula: see text] are defining relations for the group.

Duality and exact results for conductivity of 2D isotropic heterophase systems in magnetic field

Using a fact that the effective conductivity σ ˆ e of 2D random heterophase systems in the orthogonal magnetic field is transformed under some subgroup of the linear fractional group, connected with a group of linear transformations of two conserved currents, the exact values for σ ˆ e of isotropic heterophase systems are found. As known, for binary ( N = 2 ) systems a determination of exact values of both conductivities (diagonal σ e d and transverse Hall σ e t ) is possible only at equal phase concentrations and arbitrary values of partial conductivities. For heterophase ( N ⩾ 3 ) systems this method gives exact values of effective conductivities, when their partial conductivities belong to some hypersurfaces in the space of these partial conductivities and some phase concentrations are pairwise equal. In all these cases σ e does not depend on phase concentrations. The complete, 3-parametric, explicit transformation, connecting σ e in binary systems with a magnetic field and without it, is constructed.

Bounding and nonbounding finite group actions on surfaces

We consider the problem of which finite orientation-preserving group actions on closed surfaces extend to compact 3-manifolds. A solution is known for cyclic, dihedral and Abelian groups. In the present paper, we consider actions of the linear fractional groups PSL(2, p n ). Our main results imply that, for primes p≡1 mod 4 , all actions of the groups PSL(2, p n ) bound compact 3-manifolds. In particular, we show that all isometric Hurwitz and genus actions of these groups on hyperbolic surfaces bound geometrically, i.e., extend isometrically to compact hyperbolic 3-manifolds with totally geodesic boundary. On the other hand, for all primes p≡3 mod 4 there exist nonbounding actions of PSL(2, p n ).

Translations of the squares in a finite field and an infinite family of 3-designs

Let q be a prime power with q≡−1 ( mod 4) , and let F be the finite field with q elements and Q the set of nonzero squares in F. Let G= PSL(2, q) be the special linear fractional group on Ω={∞}∪F, the projective line over F, and set V={∞}∪( Q▵( Q+1)▵( Q−1)), V=Ω⧹V , where ▵ denotes the symmetric difference. First, we consider the cardinality of intersections of some translations of Q in F and show |Q∩(Q+1)∩(Q−1)|= (q−7)/8 if 2∈Q, (q−3)/8 otherwise. Next, when 2∉ Q, we determine the structure of G V=G V , the setwise stabilizer of V or V in G, and show that the design (Ω, V G) is a 3-( q+1,( q−3)/2, λ) design, where λ= (q−3)(q−5)(q−7)/64 for p≠3, (q−3)(q−5)(q−7)/(3·64) for p=3. This is a new infinite family of 3-designs.

Some maximal normal subgroups of the modular group

For each finite group G, let G denote the set of all normal subgroups of the modular group Γ = PSL2(ℤ) with quotient group isomorphic to G; since Γ is finitely generated, the number NG = |G| of such subgroups is finite. We shall be mainly concerned with the case where G is the linear fractional group PSL2(q) over the Galois field GF(q), in which case we shall write (q) and N(q) for G and NG; for q&gt;3, PSL2(q) is simple, so the elements of (q) will be maximal normal subgroups of Γ.

An Operand for a Group of Order 1512

1. The tableau on p. 105 built with the digits 0, 1,2, 3, 4, 5, 6, 7, 8 is one of a possible 1920 and was compiled in the year 1963 at about the time when [5] was published. Whether it is new or not, its construction, described below, with the help of the geometry of Study's quadric S, in projective space of 7 dimensions over the finite field GF(2), may be. One can observe the visual effects on the tableau of 1512 even permutations of the group H for which it is, as a whole, invariant. H could, in this context, be identified as the intersection of three alternating groups of degree 9 derivable from each other by using the triality on S. But H and its simple subgroup h of order 504, isomorphic to the linear fractional group LF(2, 2), occur also in a less elaborate geometry. LF(2, 2) is the group of projectivities on a 9-point line A and is extended to a group isomorphic to H by the automorphism of period 3 of GF(2) which, replacing each mark by its square, leaves three points (those with parameters 0,1, oo) of A unmoved while permuting the others in two cycles of three—a permutation unattainable by projectivities because any projectivity which leaves three points all unmoved can only be the identity.

Generators for Alternating and Symmetric Groups

Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newman's interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF(2,p) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

Groups Generated by two Parabolic Linear Fractional Transformations

We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the formwhile Chang, Jennings, and Ree (4) took them in the form

Maximal normal subgroups of the modular group

Furthermore, those of level $p are not congruence groups. It is well known that r contains infinitely many normal of finite index which are not congruence groups; see for example papers [3], [4]. However, all of these groups have the common feature that they are lattice subgroups (in Rankin's terminology) of some normal congruence group, and so are not maximal. The results of this paper imply that r contains infinitely many maximal normal of finite index which are not congruence groups, a somewhat surprising fact. The question which originally motivated this paper was the following: Which of the known simple groups have a representation as a modular quotient group, or equivalently, may be generated by two elements, one of period 2, the other of period 3? Call such a group a r-group. Then the groups LF(2, p) are certainly r-groups. However, it is not known for example when the alternating group is a r-group. In his lecture notes on Fuchsian groups [2], Macbeath poses a similar question for H-groups, and makes some remarks about the linear fractional groups which in fact form the basis of this paper, and which we are happy to acknowledge here.