Collineation groups whose order is divisible by a p-primitive divisor

1985 ◽  
Vol 24 (1) ◽  
pp. 77-88
Author(s):  
M. J. Kallaher ◽  
T. G. Ostrom
Keyword(s):  
Collineation Groups ◽  
Primitive Divisor

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals A Lucas–Lehmer approach to generalised Lebesgue–Ramanujan–Nagell equations

2021 ◽  
Author(s):  
Vandita Patel
Keyword(s):  
Computationally Efficient ◽  
Efficient Approach ◽  
Coprime Integers ◽  
Primitive Divisor

AbstractWe describe a computationally efficient approach to resolving equations of the form $$C_1x^2 + C_2 = y^n$$ C 1 x 2 + C 2 = y n in coprime integers, for fixed values of $$C_1$$ C 1 , $$C_2$$ C 2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

On Finite Groups with an Abelian Sylow Group

1962 ◽  
Vol 14 ◽  
pp. 436-450 ◽  
Author(s):  
Richard Brauer ◽  
Henry S. Leonard
Keyword(s):  
Finite Groups ◽  
Irreducible Characters ◽  
Collineation Groups ◽  
Sylow Group

We shall consider finite groups of order of g which satisfy the following condition:(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).

scholarly journals Affine planes with primitive collineation groups

1990 ◽  
Vol 128 (2) ◽  
pp. 366-383 ◽  
Author(s):  
Yutaka Hiramine
Keyword(s):  
Affine Planes ◽  
Collineation Groups

Collineation Groups of Generalized André Planes

1969 ◽  
Vol 21 ◽  
pp. 358-369 ◽  
Author(s):  
David A. Foulser
Keyword(s):  
Main Question ◽  
Translation Planes ◽  
Hall Plane ◽  
Collineation Groups

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).

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