Classification Of Primitive Permutation Groups Research Articles

On the reducibility behavior of Thue polynomials

We prove a result about reducibility behavior of Thue polynomials over the rationals that was conjectured in [7]. More precisely, we show that, apart from few explicitly given exceptions, these polynomials have only finitely many reducible integer specializations. Special cases have been proved e.g. by Müller in [7], Theorem 4.9, and Langmann [6, Satz 3.5].The proof uses ramification theory to reduce the assertion to a statement about permutation groups containing an n-cycle. This statement is finally proven with the help of the classification of primitive permutation groups containing an n-cycle (a result which rests on the classification of finite simple groups).

A classification of primitive permutation groups with finite stabilizers

We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher–O'Nan–Scott Theorem to all primitive permutation groups with finite point stabilizers.

On Permutation Groups of Degree a Product of Two Prime-Powers

We determine finite simple groups which have a subgroup of index with exactly two distinct prime divisors. Then from this we derive a classification of primitive permutation groups of degree a product of two prime-powers.

Some Constant Weight Codes from Primitive Permutation Groups

In recent years the detailed study of the construction of constant weight codes has been extended from length at most 28 to lengths less than 64. Andries Brouwer maintains web pages with tables of the best known constant weight codes of these lengths. In many cases the codes have more codewords than the best code in the literature, and are not particularly easy to improve. Many of the codes are constructed using a specified permutation group as automorphism group. The groups used include cyclic, quasi-cyclic, affine general linear groups etc. sometimes with fixed points. The precise rationale for the choice of groups is not clear.In this paper the choice of groups is made systematic by the use of the classification of primitive permutation groups. Together with several improved techniques for finding a maximum clique, this has led to the construction of 39 improved constant weight codes.

Nonorientable regular embeddings of graphs of order pq

In [J. Algeb. Combin.19(2004), 123–141], Du et al. classified the orientable regular embeddings of connected simple graphs of order pq for any two primes p and q. In this paper, we shall classify the nonorientable regular embeddings of these graphs, where p ≠ q. Our classification depends on the classification of primitive permutation groups of degree p and degree pq but is independent of the classification of the arc-transitive graphs of order pq.

On 2-arc-transitive representations of the groups of fourth-power-free order

A complete classification of 2 -arc-transitive representations of primitive or bi-primitive groups of fourth-power-free order is given. The list consists of the following graphs: K n , K n , n − n K 2 , K r , r with r an odd prime, Peterson graph O 2 , incidence and non-incidence graphs H D ( 11 , 5 , 2 ) and H D ( 11 , 6 , 2 ) of the Hadamard design on 11 points, some explicit coset graphs of two-dimensional projective groups PSL ( 2 , p l ) with 1 ≤ l ≤ 2 and of Janko simple group J 1 , and the standard double cover of these coset graphs. Moreover, a classification of primitive permutation groups of fourth-power-free order is given.

A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p ≠ q and it is also independent of the classification of the arc-transitive graphs of order pq for p ≠ q.