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Let G be a finite subgroup of PGL 3 (C) , and let σ be the generator of Gal (C / R) . We say that G has a real field of moduli if σ G and G are PGL 3 (C) -conjugates. Furthermore, we say that R is a field of definition for G or that G is definable over R if G is PGL 3 (C) -conjugate to some G ´ ⊂ P G L 3 (R) . In this situation, we call G ´ a model for G over R . On the other hand, if G has a real field of moduli but is not definable over R , then we call G pseudo-real. In this paper, we first show that any finite cyclic subgroup G = Z / n Z in PGL 3 (C) has a real field of moduli and we provide a necessary and sufficient condition for G = Z / n Z to be definable over R ; see Theorems 2.1, 2.2, and 2.3. We also prove that any dihedral group D 2 n with n ≥ 3 in PGL 3 (C) is definable over R ; see Theorem 2.4. Furthermore, we study all other classes of finite subgroups of PGL 3 (C) , and show that all of them except A 4 n , A 5 n and S 4 n are pseudo-real; see Theorems 2.5 and 2.6. Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry; see Theorem 2.7 and Example 2.8. As a result, we can say that if G is definable over R , then its Jordan constant J (G) = 1, 2, 3, 6 or 60. [ABSTRACT FROM AUTHOR]

Let F q (x) be the field of rational functions over F q and treat PGL (2 , F q) as the group of degree one rational functions in F q (x) equipped with composition. PGL (2 , F q) acts on F q (x) from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup H of PGL (2 , F q) is the stabilizer of some rational function π H (x) ∈ F q (x) with deg π H = | H | under this action, where π H (x) is uniquely determined by H up to a left composition by an element of PGL (2 , F q). In this article, we determine the rational function π H (x) explicitly for every H < PGL (2 , F q). [ABSTRACT FROM AUTHOR]

Let C be the field of complex numbers and n and k natural numbers satisfying k ≥ 2 and n ⩾ k + 3. Let P n k (C) be the subset of nth symmetric product (P k (C)) (n) of P k (C) with itself consisting of n-point sets in general position and let M n k (C) = P n k (C) / PGL k + 1 (C). We describe the dimension of the points in M n k (C) which correspond to n-point sets whose automorphism group contains a subgroup of prime order. From this result, we prove that the points in M n k (C) which correspond to n-point sets with nontrivial automorphism group form a proper subset M n k (C). [ABSTRACT FROM AUTHOR]

We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks, and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are “reflective” versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. [ABSTRACT FROM AUTHOR]

A maniplex of rank n is a combinatorial object that generalises the notion of a rank n abstract polytope. A maniplex with the highest possible degree of symmetry is called regular. In this paper we prove that there is a rank 4 regular maniplex with automorphism group PSL 2 (q) for infinitely many prime powers q , and that no regular maniplex of rank n > 4 exists that has PSL 2 (q) as its full automorphism group. [ABSTRACT FROM AUTHOR]

We study the orbit of R̂ under the Möbius action of the Bianchi group PSL2(OK) on Ĉ, where OK is the ring of integers of an imaginary quadratic field K. The orbit SK, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of K. Wegive a simple geometric characterisation of certain subsets of SK generalizing Apollonian circle packings, and show that SK, considered with orientations, is a disjoint union of all primitive integral such K-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called KApollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings. [ABSTRACT FROM AUTHOR]

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group S T + (n , F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL (n , F) , where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL (p − 1 , Q) is minimal, where Q is the field of rationals equipped with the p -adic topology; (3) SL (p − 1 , Q (i)) is minimal, where Q (i) ⊂ C is the Gaussian rational field. [ABSTRACT FROM AUTHOR]

Here is a simplified proof that every sharply transitive subset of PGL2(K) is a coset of a subgroup, for every finite field K. [ABSTRACT FROM AUTHOR]

Letandbe vector spaces over division rings. Supposeis finite and not less than. Consider a mappingwith the following property: for every, there issuch that. Our first result states thatis a strong semilinear embedding ifis non-constant and the dimension of the subspace ofspanned byis not greater than. We present examples showing that these conditions cannot be omitted. Denote bythe projective space associated withand consider the mappingwith the following property: for every, there issuch that. By the second result,is induced by a strong semilinear embedding ofinifis non-constant and its image is contained in a subspace ofwhose dimension is not greater than, we also require thatis a field. [ABSTRACT FROM PUBLISHER]

Abstract: We prove a new upper bound on the essential p-dimension of the projective linear group . [Copyright &y& Elsevier]

Abstract: Let , , q a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group G with . In particular, we show that G cannot act as a group of automorphisms on any Steiner quadruple system for . [Copyright &y& Elsevier]

Abstract: Let be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space over k, and we compute the generating function for the numbers of -orbits of these n-sets. For we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients. [Copyright &y& Elsevier]

Abstract: Let be a projective plane, and let and PSL(2, q)⩽ G ⩽PΓL(2, q) with q >3. If G acts point-transitively on , then q =7 and is of order 2. [Copyright &y& Elsevier]

We study 4-restricted min-wise independent sets of permutations. In particular, we prove that the groups PGL(2, q) (for any prime power q) and PGL(3, q) (for any odd prime power q), permuting points of the projective line and the projective plane over the field Fq, respectively, satisfy this property. We also show that (1) for any 4-restricted min-wise independent set G _CSn, its cardinality is at least 2n-2, and (2) for any sufficiently large n, there exists a 4-restricted min-wise independent set G _CSn of size n³ + o(n³). The last two results improve previous bounds found by Itoh, Takei, and Tarui. [ABSTRACT FROM AUTHOR]

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at simple groups and prove the following theorem:Let G=PSU(3,q) with q even and G acts line-transitively on a finite linear space S. Then S is one of the following cases:(i)A projective plane;(ii)A regular linear space with parameters (b,v,r,k)=(q2(q2−q+1),q3+1,q2−q+1,q+1). This is called the Hermitian unitary design. [Copyright &y& Elsevier]

This article is a contribution to the study of linear spaces admitting a line-transitive automorphism group. We classify such linear spaces where PSL(2,q), q>3 acts line transitively. We prove that the only cases which arise are projective planes, a Bose–Witt–Shrikhande linear space and one more space admitting PSL(2,26) as a line-transitive automorphism group. [Copyright &y& Elsevier]

This paper is devoted to the classification of flag-transitive 2- (v , k , 2) designs. We show that apart from two known symmetric 2- (16 , 6 , 2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2- (v , k , 2) design is primitive of affine or almost simple type. Moreover, we classify the 2- (v , k , 2) designs admitting a flag transitive almost simple group G with socle PSL (n , q) for some n ≥ 3. Alongside this analysis we give a construction for a flag-transitive 2- (v , k − 1 , k − 2) design from a given flag-transitive 2- (v , k , 1) design which induces a 2-transitive action on a line. Taking the design of points and lines of the projective space PG (n − 1 , 3) as input to this construction yields a G -flag-transitive 2- (v , 3 , 2) design where G has socle PSL (n , 3) and v = (3 n − 1) / 2. Apart from these designs, our classification yields exactly one other example, namely the complement of the Fano plane. [ABSTRACT FROM AUTHOR]