Showing papers on "Symmetric group published in 1972"

The automorphism group of an extraspecial $p$-group

Abstract: 1. Let p be a prime. The finite p-group P is called special if either (i) P is elementary abelian or (ii) the center, commutator subgroup and Frattini subgroup of P all coincide and are elementary abelian. A nonabelian special p-group whose center has order p is called an extraspecial p-group. It is possible to give a uniform treatment of the subject of automorphisms for all the possible isomorphism types of extraspecial p-groups and so some cases that are more or less known are included here. The result when p is odd and P has exponent p leads to an interesting subgroup of the symplectic group Sp (2n, q), q a power of p, n > 1. This subgroup is the semidirect product of Sp (2n — 2, q) and a normal special p-group of order q~ whose center has order q.

Enumeration of Permutational Isomerization Reactions

Abstract: Rearrangement reactions of certain stereochemically nonrigid molecules may be described by permutation operations. The usefulness of these permutation operations in solving quantum mechanical and stereo‐chemical problems has already been demonstrated. In this paper it is shown how the point group of a molecule renders certain permutations equivalent. This equivalence is used to generate classes of equivalent permutations, i.e., classes of equivalent isomerization reactions. Two types of equivalent reactions are defined: indistinguishable permutational isomerization reactions and nondifferentiable permutational isomerization reactions. The classes of equivalent permutations generated are shown to be conjugacy classes and double cosets, respectively, of the symmetric group without respect to a subgroup. Formulas are provided which enumerate these classes. Nonrigid trigonal bipyramidal and octahedral molecules are treated as examples.

Transitive Permutation Groups of Prime Degree

Abstract: This paper is intended as a survey of what is now known about transitive permutation groups of prime degree. The topic arises from early work on the theory of equations. Over 200 years ago Lagrange was led to an interest in irreducible polynomial equations of prime degree by showing1 that if every such equation were soluble in terms of root-extraction then polynomial equations of arbitrary degree would be. Even after Abel and Galois had shown that such solutions are impossible in general, Galois still devoted a good proportion of his work to equations of prime degree. It was of course he who emphasised the groups involved. Several 19th century mathematicians, notably Mathieu and Jordan, continued the work of Galois and provided foundations for the rich material that has been published since 1900. At present the problems concerning groups of prime degree remain near the centre of permutation group theory, retaining their interest partly as tests of the power and scope of techniques of finite group theory, partly as being typical of a range of similar problems concerning groups of degrees kp (with k < p) and p m where p is prime.

On certain maximal subgroups of symmetric and alternating groups

Abstract: This paper deals with the problem of describing the maximal subgroups of symmetric and alternating groups. Certain primitive groups which are not multiply transitive are considered, and it is shown that among these there are infinite series of groups which are maximal in symmetric and alternating groups. Figures: 3. Bibliography: 15 items.

On some problems of a statistical group theory VII

Abstract: 1. By statistical group-theory we mean the study of those properties of certain complexes of a “large” group which are shared by “most” of these complexes. The group considered in this paper will be &, the symmetric group of n letters; its group-elements will be denoted by P. The complexes considered here will be simply the elements P of 8, ; the property in question will be the group-order O(P) of P. As to this LANDAU proved (see [a]) for

Distance Properties of Group Codes for the Gaussian Channel

Abstract: Some distance properties of group codes for the Gaussian channel introduced by Slepian (1968) are examined. The concept of a full homogeneous component is introduced and optimal vectors for such group representations are found. The results are applied to the symmetric and Mathieu groups which are found to yield exceptional simplex-like codes with code size larger than the corresponding simplex code. Finally, a theorem on the representation of a doubly transitive permutation group is used to solve the optimal vector problem for the irreducible representation of dimension $n - 1$ of the symmetric group of degree n.

The Symmetric Group and the Gel'fand Basis of U(3). Generalizations of the Dirac Identity

Abstract: It is shown that the symmetrization of N particle states by means of the orthogonal units of the algebra of the symmetric group SN yields the Gel'fand basis states of the irreducible representations of U(3). The existence of generalizations of the Dirac identity is demonstrated, and a connection between the symmetrized two‐ and three‐body exchange operators and the invariants of U(3) is established.

On the fundamental group and homotopy type of open 3-manifolds

Abstract: It is shown that a finitely-generated subgroup of the fundamental group of a 3- manifold is the fundamental group of a compact 3-manifold if the subgroup can be suitably approximated by a finitely-presented group. This result is then applied to study certain subgroups of 3-manifold fundamental groups: those subgroups which are finitely-generated and have finite abelianizations. In many cases, these subgroups are shown to be isomorphic to fundamental groups of closed 3-manifolds. Corollaries concerning the embedding of spheres and cells in 3- space are given. A necessary and sufficient condition is given in order that an open 3-manifold with no 2-sided projective planes should be homotopy-equivalent to a compact 3-manifold.

On cyclic maps

Abstract: Throughout X will denote a connected, finite C.W. complex. Let G be a subgroup of ∑ n , the symmetric group, which acts transitively on the Cartesian product, X n , of the space X . A map f : X n → X is G -symmetric if it commutes with the action of G . If x 0 e X is a base point let i : X → X n denote the inclusion, i ( x ) = ( x , x 0 , …, x 0 ). In ((6); (7); (11)–(13)) the following problem is posed: if X is an orientable topological n -manifold, what is the set of integers which may be obtained as the degree of( f . i ) where f is a G -symmetric map? The degree of( f . i ) is called the James number of f . If G = ∑ n ( G = Z n ) a G -symmetric map will be called a symmetric map (a cyclic map). If X = S n , the n-sphere, this problem has been studied in ((6)–(8), (11)–(13))

The n-electron problem and matrices representing the symmetric groups

Abstract: The application of the representation theory of symmetric groups to the solution of quantum chemical problems is discussed. This is done using matrices which do not represent Sn in the normal sense, but which are much simpler to calculate than the Yamanouchi-Kotani representation. A complete discussion of the calculation of these matrices is given.

On the radical of the group algebra of a $p$-group over a modular field

Abstract: Let G be a finite p-group, K be the field of integers modulop, KG be the group algebra of G over K and N be the radical of KG. By using the fact that the annihilator, A(N), of N is one dimensional, we characterize the elements of A (N2). We also present relationships among the cardinality of A(N2), the number of maximal subgroups in G and the number of conjugate classes in G. Theorems concerning the Frattini subalgebra of N and the existence of an outer automorphism of N are also proved.

BLT-sets admitting the symmetric group S4

Abstract: We describe a set of subgroups isomorphic to S4 in PGO(5, q), q ?? 5(6), and prove that they belong to exactly two different conjugacy classes. As an application we use a representative group in each of the conjugacy classes to construct a number of BLT-sets of Q(4, q), some of which were previously found by computer in [5] (see also [4]), others of which are new.

Subgroups dual to dimension subgroups

Abstract: Associated with, the powers of the augmentation ideal are the dimension subgroups. In the integral group ring case, they have long been conjectured to be the terms of the lower central series. This paper investigates the subgroups associated with the chain of ideals dual to the chain of powers of the augmentation ideal. The study is reduced to the case of the modular group rings of p -groups. The subgroups are calculated for Abelian p -groups, p odd. They appear in the upper central series of wreath products and provide a new criterion for the nilpotence of an arbitrary wreath product. The nilpotence class of wreath products is considered here as well; calculations and bounds are given; in particular, a new method of computing the class of the Sylow p -subgroups of the symmetric group arises.

On the Greatest Order of an Element of the Symmetric Group

Abstract: Let f(n) denote the greatest order of a permutation in the symmetric group S". Since Sn is isomorphic to a subgroup of S,+1, it follows that f(n) ? f(n + 1), and so f increases monotonically. Using the prime number theorem, Landau [2, p. 225] showed that logf(n) is asymptotic to In log n, and Shah [3] has slightly strengthened this result. In this note I give an elementary proof that f(n) grows faster than any power of n. All lower case latin letters stand for integers. The greatest common divisor and least common multiple of a1,, ak are denoted (a1, *,ak) and [a1, ak] respectively.

A note on a theorem of Alexander

Abstract: represented link ("nudo coloreado'') (L,ωn) is a tame link L in S3 together with a transitive representation ωn of π1(S3−L,∗) into the symmetric group Sn; it can easily be pictured in the plane by a regular knot projection and a labelling of the overpasses by elements of Sn. With (L,ωn) there is canonically associated an n-fold covering space C(L,ωn) of S3 branched over L. J. W. Alexander showed in 1920 that any connected orientable closed 3-manifold M is C(L,ωn) for some represented link; in addition he asserted that we may have branching index ≤2 everywhere [see R. H. Fox, Topology of 3-manifolds and related topics (Proc. Univ. Georgia Inst., 1961), pp. 213–216, Prentice-Hall, Englewood Cliffs, N.J., 1962; MR0140116 (25 #3539)]. In the present paper a new proof of this assertion is given in a slightly stronger version; in fact for given M the author exhibits an n∈N and a represented link (L˜,ω˜n) such that (i) M=C(L˜,ω˜n) and moreover (ii) the label of each overpass is a permutation of (1,⋯,n) which only permutes some pair of these numbers. The method of proof consists in applying to some (L,ωn) satisfying (i) a series of modifications which leave the topological type of the covering space invariant but gradually change (L,ωn) to satisfy (ii) in the end. The usefulness of this method is illustrated by determining the topological type of some C(L,ωn); in particular, the simply connected 3-manifolds proposed by R. H. Fox [op. cit.] as possible counterexamples to the Poincare conjecture are shown to be S3.

Normal subgroups of the multiplicative group of a ring

Abstract: In the paper we study the construction of K-linearly independent normal subgroups of the multiplicative group U(R) of an algebra R over a ring K of characteristic pm and their ordering inside the groups U(R). We also describe the group rings whose multiplicative group contains a K-linearly independent normal subgroup not belonging to the center and containing an element of finite order. We also study abelian normal subgroups of the multiplicative group of the ring of a p-group over a ring of characteristic p. Bibliography: 11 items.

Some applications of the representation theory of finite groups. A partial reduction methof

Abstract: In this thesis we study the representation theory of finite groups and more specifically some aspects of the theory of characters. The technique of symmetrization and/or antisymmetrization of Kronecker powers of representations, which is well-known for the general linear group is applied here to representations of arbitrary finite groups. In Chapter 2 we show how one can partially reduce the nth- Kronecker power of an irreducible representation of a finite group. In the reduction the characters of the irreducible representations of the symmetric group play an important role. In order to apply the reduction formulae some properties of the characters of the symmetric group are derived. In Chapter 3 we discuss various theorems concerning the number of roots of certain equations in finite groups. We show that in many cases one can get information on the number of roots of these equations from the knowledge of the existence of certain representations. Furthermore various generalizations are given of a Theorem of Frobenius and Schur concerning some group averages. In Chapter 4 analogous properties are derived on the number of classes which are invariant for substitutions of the group elements. Some theorems of Chapter 3 suggest a kind of duality betweenthe classes and the irreducible representations of a finite group. In Chapter 5 some other examples of this duality are given. Among others a theorem dual to a well-known theorem of Cauchy is derived. The considerations of the previous chapters are applied in Chapter 6 to the problem of the phases of Clebsch-Gordan coefficients. These coefficients have simple symmetry properties if an equality holds between two group averages. These group averages are of the form as studied in Chapter 3. Groups for which the above equality holds for all irreducible representations we shall call simple phase groups or S.P. groups. Derome showed by direct verification of the equality that e.g. the group SU(3) which plays an important role in elementary particle physics is an S.P. group. On the other hand it appeared that the groups SU(4) and the symmetric group S6 are non-S.P. groups. In Chapter 6 we give more examples of groups which are not S.P. For instance the K-metacyclic groups (which form an infinite series) appear to be non-S.P. groups. The considerations of Chapter 6 are continued in Chapter 7 where we derive several criteria for non-S.P. groups. It is easier to apply these criteria than the definition of S.P. groups itself, which may lead to long and tedious calculations. Furthermore we make clear that the property of being non-S.P. of a group has much resemblance to the property of having representations of type 2. Type 2 representations are representations with real character, but which cannot be transformed to real form. We also derive criteria for type 2 representations which are quite similar to the criteria for non-S.P. groups. In particular we give a proof of a theorem of Frobenius and Schur on the existence of type 2 representations for a certain kind of groups and also its analogue for non-S.P. groups. Examples of groups which are covered by the various criteria are given in the last chapter. We also give a procedure by which one can construct groups having type 2 representations or being non-b.P., starting from groups which do not have these properties. Lastly we make some remarks on the application of our theorems to the calculation of character tables. In the appendices we give a compilation of the character tables of the groups which have been used as an illustration of the various theorems. In this thesis it is assumed that the reader is familiar with the representation theory of groups as can be found in any textbook. As far as our notation concerns we mostly stick to the notations of refs. [1] and [2].