Note on primitive permutation groups and a diophantine equation

Journal of the London Mathematical Society, 1973

In a series of papers [3, 4 and 5] on insoluble (transitive) permutation groups of degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from a small number of exceptions, such a group must be at least quadruply transitive. One of the results which he uses is that an insoluble group of degree p = 2q +1 which is not doubly primitive must be isomorphic to PSL (3, 2) with p = 7. This result is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extend this proof to give the following result: a doubly transitive group of degree 2q + l, where q is prime, which is not doubly primitive, is either sharply doubly transitive or a group of automorphisms of a block design with A = 1 and k = 3. Our notation for the parameters of a block design, v, b, k, r, X, is standard; see [9]. In this paper we shall prove two results about doubly transitive but not doubly primitive groups which resemble the two results mentioned above.

J. Austral. Math. Soc. Ser. …, 1988

1989

Journal of Number Theory, 2014

This note presents corrections to the paper by Y. Wang and T. Wang [2]. The unique theorem given in that paper states that for any odd integer n > 1, nx 2 + 2 2m = y n has no positive integer solution (x, y, m) with gcd(x, y) = 1.

Journal of Algebra, 2017

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition to the previous results and hierarchy, we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems.

Journal of the Australian Mathematical Society, 1988

2011

Abstract A classification is given of rank 3 group actions which are quasiprimitive but not primitive. There are two infinite families and a finite number of individual imprimitive examples. When combined with earlier work of Bannai, Kantor, Liebler, Liebeck and Saxl, this yields a classification of all quasiprimitive rank 3 permutation groups.

Journal of Number Theory, 2005

Let D > 2 be a positive integer, and let p be an odd prime not dividing D. In this paper, using the deep result of Bilu, Hanrot and Voutier (i.e., the existence of primitive prime factors of Lucas and Lehmer sequences), by computing Jacobi's symbols and using elementary arguments, we prove that: if (D, p) = (4, 5), (2, 5), then the diophantine equation x 2 + D m = p n has at most two positive integer solutions (x, m, n). Moreover, both x 2 + 4 m = 5 n and x 2 + 2 m = 5 n have exactly three positive integer solutions (x, m, n).

International Journal of Discrete Mathematics

The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x 2 +y 2 =z 2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n<m are mutually Prime numbers of different parity, and this happens in cases where, for example, n=4, and m is a Prime number ending in 3 or 7. In such cases, the hypotenuse z=m 2 +n 2 is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.

Transactions of the American Mathematical Society, 2004

Let G and X be transitive permutation groups on a set Ω such that G is a normal subgroup of X. The overgroup X induces a natural action on the set Orbl(G, Ω) of non-trivial orbitals of G on Ω. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples (G, X, Ω) where X fixes no elements of Orbl(G, Ω); such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples (G, X, Ω, P) where P is a partition of Orbl(G, Ω) such that X is transitive on P; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple (G, X, Ω) in a TOD (G, X, Ω, P) is exceptional; conversely if an exceptional triple (G, X, Ω) is such that X/G is cyclic of prime-power order, then there exists a partition P of Orbl(G, Ω) such that (G, X, Ω, P) is a TOD. This paper characterizes TODs (G, X, Ω, P) such that X Ω is primitive and X/G is cyclic of primepower order. An application is given to the classification of self-complementary vertex-transitive graphs.