Digraph Representations of 2-closed Permutation Groups with a Normal Regular Cyclic Subgroup

The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.

Multiple holomorphs of finite p-groups of class two

A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970's. It is shown that a connected arc transitive circulant ? of order n is one of the following: a complete graph Kn, a lexicographic product $\Sigma [{\bar K}_b]$ , a deleted lexicographic product $\Sigma [{\bar K}_b] - b\Sigma$ , where ? is a smaller arc transitive circulant, or ? is a normal circulant, that is, Auta? has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.

The process of imbedding a group in a larger group of some prescribed type has been one of the most useful tools in the investigation of properties of groups. The three principal types of representation of groups, each with its particular field of usefulness, are the following: 1. Permutation groups. 2. Monomial groups. 3. Linear or matrix representations of groups. These three types of representation correspond to an imbedding of the group in the following groups: 1. The symmetric group. 2. The complete monomial group. 3. The full linear group. The symmetric group and the full linear group have both been exhaustively investigated and many of their principal properties are known. A similar study does not seem to exist for the complete monomial group. Such a general theory seems particularly desirable in view of the numerous recent investigations on finite groups in which the monomial representations are used in one form or another to obtain deep-lying theorems on the properties of such groups. The present paper is an attempt to fill this lacuna. In this paper the monomial group or symmetry is taken in the most general sense(') where one considers all permutations of a certain finite number of variables, each variable being multiplied also by some element of a fixed arbitrary group H. In the first chapter the simplest properties such as transformation, normal form, centralizer, etc., are discussed. Some of the auxiliary theorems appear to have independent interest. One finds that the symmetry contains a normal subgroup, the basis group, consisting of all those elements which do not permute the variables. The symmetry splits over the basis group with a group isomorphic to the symmetric group as one representative group. A complete solution of the problem of finding all representative groups in this splitting of the symmetry is given. This result is of interest since it gives a general idea of the solution of the splitting problem in a fairly complicated case. In the second chapter all normal subgroups of the symmetry are deter-

On solvable factors of almost simple groups

We give a useful description of the permutation groups containing a regular abelian Hall subgroup.

W. Burnside [3, p. 343] showed that a cyclic group of order pm (p prime, m > l) cannot occur as a regular subgroup of a simply transitive primitive group. (For definitions and notation see [9].) Groups which are contained regularly in a primitive group G only when G is doubly transitive are therefore called B-groups [9, p. 64].

Let [Formula: see text] be a finite group. There is a natural Galois correspondence between the permutation groups containing [Formula: see text] as a regular subgroup, and the Schur rings (S-rings) over [Formula: see text]. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group [Formula: see text] is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

In this article we prove that there is only one symmetric transversal design STD4[12;3] up to isomorphism. We also show that the order of the full automorphism group of STD4[12; 3] is 25· 33 and Aut STD4[12;3] has a regular subgroup as a permutation group on the point set. We used a computer for our research.

CHAPTER IV - Applications of the Theory of Characters

On graph-restrictive permutation groups

A transitive permutation group is called semiprimitive if each of its normal subgroups is either semiregular or transitive. The class of semiprimitive groups properly includes primitive groups, quasiprimitive groups and innately transitive groups. The latter three classes of rank 3 permutation groups have been classified, making significant progress towards solving the long-standing problem of classifying permutation groups of rank 3. In this paper, we complete the classification of finite semiprimitive groups of rank 3, building on the recent work of Huang, Li and Zhu. Examples include Schur coverings of certain almost simple 2-transitive groups and three exceptional small groups.

This chapter opens Part Two, devoted to the study of transformation theory, with some additional topics in group theory that arise in musical applications. Transformation groups on finite spaces may be regarded as permutation groups; permutation groups on pitch-class space include not only the groups of transpositions and inversions but also the multiplication group, the affine group, and the symmetric group. Another musical illustration of permutations involves the rearrangement of lines in invertible counterpoint. The structure of a finite group may be represented in the form of a group table or a Cayley diagram (a kind of graph). Other concepts discussed include homomorphisms and isomorphisms of groups, direct-product groups, normal subgroups, and quotient groups. Groups underlie many examples of symmetry in music, as formalized through the study of equivalence relations, orbits, and stabilizers.

Let Γ be a group of order mp where p is prime and p>m. We give a strategy to enumerate the regular subgroups of Perm(Γ) normalized by the left representation λ(Γ) of Γ. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions L∕K with Γ= Gal(L∕K). We prove that every such regular subgroup is contained in the normalizer in Perm(Γ) of the p-Sylow subgroup of λ(Γ). This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order p(p−1), where p is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.