Showing papers on "Symmetric group published in 1970"

Computational methods in the study of permutation groups

Abstract: Publisher Summary This chapter describes the computational methods in the study of permutation groups. It is useful to extend the determination of the primitive groups of low degree. With advances in group theory and the availability of electronic computers for routine calculations, it is feasible to carry out the determination as far as degree 30, and probably farther. The procedure combines the use of a computer with more conventional techniques. The primitive groups of a given degree n are determined. It is assumed that the primitive groups of degree less than n have already been determined. As a minimal normal subgroup of a primitive group is transitive and is a direct product of isomorphic simple groups, the first step is to take each of the known simple groups H, including the groups of prime order, determine the transitive groups M of degree n isomorphic to the direct product of one, or more copies of H, and then for each such group M find the primitive groups containing M as a normal subgroup.

Subgroups of Fuchsian Groups and Finite Permutation Groups

Abstract: We then say F has signature (g;mi,m2,...,mr;s;t) (1) The integers mu m2,... mr are called the periods of F. We note the following facts about this presentation. Every elliptic element of F is conjugate to a power of one of the Xj{\ < j ^ r), every parabolic element of T is conjugate to a power of one of the pk(l < k < s) and every hyperbolic boundary element of T is conjugate to a power of one of the ht(l ^ / ^ t). Moreover no nontrivial power of one of the generators can be conjugate to a power of another generator. These well-known facts follow from the existence of a fundamental region for T with the property that the generators are the elements which map the region onto a full neighbour [1, 2]. The problem we consider here is to determine necessary and sufficient conditions for a subgroup of finite index in F to have some given signature. In §3 we give an application to finite permutation groups.

The number of conjugacy classes in a finite group

Abstract: This was proved by Sah [6], in the case in which G/N is abelian. We also show that equality in (2) implies H is normal. Ernest's proofs of (1) and (2) are character-theoretic, in w 1, we give elementary proofs of (1), (2), and (3). In w 2, a computation of Schur [7] giving the number of inequivalent irreducible projective representations of a finite group with a given factor set is adapted to give the number of distinct irreducible components in the character induced by an invariant irreducible character of a normal subgroup. In w 3, this is applied to give a character-theoretic proof of (3) with a character-theoretic criterion for equality.

Metacyclic invariants of knots and links

Abstract: To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o 1, o 2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.

States and automorphism groups of operator algebras

Abstract: Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms As a result we obtain a “noncommutative” ergodic theorem The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian

The characters of the Weyl group E8

Abstract: This chapter discusses the characters of the Weyl group E 8 . The group F of order 192.10! = 2 14 3 5 5 2 7 = 696,729,600 whose 112 absolutely irreducible characters (all rational) are described in this chapter is isomorphic to the Weyl group E 8 . The group F itself is described by Coxeter as the eight-dimensional group 3 [4,2,1] of symmetries of Gosset's semi-regular polytope 4 21 , and it is the largest of the irreducible finite groups generated by reflections. Its factor group A = F / C with respect to its center C = { I , − I } is the orthogonal group of half the order investigated by Hamill as a collineation group and by Edge as the group A of automorphisms of the nonsingular quadric consisting of 135 points of a finite projective space. The simple group denoted FH (8, 2) by Dickson is a subgroup A + of index 2 in A = F / C .

Symmetry Adaptation to Sequences of Finite Groups

Abstract: Publisher Summary This chapter provides an introduction to symmetry adaptation and the assignment of group theoretical quantum numbers For symmetry adaptation with respect to a finite group, the matric basis of the Frobenius (group) algebra may be employed Wedderburn's construction of a matric basis is reviewed For symmetry adaptation with respect to a group sequence, a special matric basis called the sequence adapted matric basis may be employed The chapter also presents the modification of Wedderburn's method to yield a constructive proof of the existence of a sequence adapted matric basis Developments of the properties of the sequence adapted matric basis are shown The sequence adapted matric basis for special types of group sequences is considered; formulas for the corresponding sequence adapted irreducible representation matrix elements are presented Further, two methods for obtaining a sequence adapted irreducible representation from one that is not sequence adapted is provided followed by a description of application to the symmetric group The chapter also presents examples from the symmetric group

On a programme for the determination of the automorphism group of a finite group

Abstract: This chapter describes a program for the determination of the automorphism group of a finite group. The program A for the determination of the automorphism group A ( G ) of a finite group G is part of a system of programs for the investigation of finite groups implemented on an Electrologica X1 at the Rechenzentrum der Universitat Kiel. A detailed description of A has been given in Numerische Mathematik . The program A makes use of information about the lattice of subgroups of the group G provided by a program F . In the working of program A , a system of generators and defining relations of G is determined. It is used later to decide whether a mapping from G onto G is a homomorphism. If G does not possess generators, then generators and defining relations must be provided as input by the user of the program.

Commuting‐Operator Approach to Group Representation Theory

Abstract: A study is given of the class‐sum‐operator approach to the representation theory of finite groups, the group D3h being specifically studied. The class sum approach is shown to simplify the decomposition of Kronecker products. By using the class‐sum‐operator approach, it is shown that the ``indirect'' group‐projection operators of Lowdin can be used in finite group theory, where they lead to useful factorizations of the finite group‐projection operators. It is also shown that tensor operators of certain symmetry types can be constructed within the group itself, and may be used analogously to the usual operator equivalents of crystal field theory.

The automorphism group of a finite metacyclic $p$-group

Abstract: In this paper it is shown that if G is a finite nonAbelian metacyclic £-group, py^t, then the order of G divides the order of the automorphism group of G. It is well known that if G is a finite noncyclic Abelian p-group of order greater than p2, then the order \G\ of G divides the order of the automorphism group A(G) of G. This result has recently been extended to other classes of finite ^-groups [l], [6]. We recall that a group G is said to be metacyclic if G possesses a cyclic normal subgroup K such that G/K is also cyclic. The purpose of this paper is to show that | G\ divides | ^4(G)| if G is a finite noncyclic metacyclic p-group of order greater than p2, p9^2. The following notation is used: G is a finite p-group where p is a prime; class G denotes the nilpotency class of G; Gn is the wth element in the descending central series of G; Z(G) denotes the center of G (or Z, if no ambiguity is possible); 77:SG means 77 is a subgroup of G, [G:77] denotes the index of 77 in G and 77

A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

Abstract: The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬx containing π, where ℬx is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons [1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky [4]. A proof of the general case, based on [1] and results due to Plemmons and West [3], is also given in the preceding note by Plemmons and Schein [2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.

Some combinatorial and symbol manipulation programs in group theory

Abstract: Publisher Summary This chapter discusses some combinatorial and symbol manipulation programs in group theory. A program has been written that determines the generators and relations of all the subgroups of a finite soluble group. The program finds the subgroups using the same method as Neubuser with the difference that as a subgroup of order d is found the generators and relations of all possible groups of order d are checked through to find a set of generators and relations for the subgroup. As the program is restricted to groups of order less than 400 by machine considerations, all the possible groups of order less than 200 need to be known. The program reads the generators and relations of the given group and finds a faithful permutation representation using coset enumeration.

Groups of Exponent 4 as Automorphism Groups.

Abstract: If a finite group A of exponent 4 is the automorphism group of a torsionfree (abelian) group G, then A must be a subdirect product of cyclic groups C2 and C4 and of quaternion groups Qs (see [2, 4]); A is therefore contained in the variety V(Qs) generated by the quaternion group. Having dealt completely in [3] with the case of finite (or countable) abelian automorphism groups, we now turn to non-abelian groups (which, as we have mentioned before ([3], p. 34), must be of exponent 4 or 12). In this paper we confine ourselves to proving that for every n>2 the free n-generator group A, of V(Qs) is, in fact, the automorphism group of a suitably chosen torsionfree group 1. We also give at the end of the paper an example (with n = 3) of an epimorphic image of Aa of order 27 that cannot be an automorphism group, although it still satisfies the condition N3 of [2]: all its elements of order 2 lie in its centre. This group fails to satisfy the necessary condition N5 of [1] and cannot be a subdirect product of the kind described above.

A computational method for determining the automorphism group of a finite solvable group

Abstract: Publisher Summary This chapter discusses a computational method for determining the automorphism group of a finite solvable group. Many problems in the theory of finite groups, especially of the extension theory, depend on the knowledge of the structure of the automorphism group A ( G ) of a finite group G . A computer program for determining A ( G ) of a finite group G has been described in the chapter. With a view to the computational construction of A ( G ) it is profitable to develop systematically methods for determining A ( G ) by composition of allowable automorphisms of special subgroups of G .