Showing papers on "Symmetric group published in 1976"

The weyl group of a graded lie algebra

Abstract: The action of the group G0 of fixed points of a semisimple automorphism θ of a reductive algebraic group G on an eigenspace V of this automorphism in the Lie algebra g of the group G is considered. The linear groups which are obtained in this manner are called θ-groups in this paper; they have certain properties which are analogous to properties of the adjoint group. In particular, the notions of Cartan subgroup and Weyl group can be introduced for θ-groups. It is shown that the Weyl group is generated by complex reflections; from this it follows that the algebra of invariants of any θ-group is free.Bibliography: 30 titles.

Higher indices of group representations

Abstract: The nth order index of an irreducible representation of a semisimple compact Lie group, n a nonnegative even integer, is defined as the sum of nth powers of the magnitudes of the weights of the representation. It is shown, in many situations, to have additivity properties similar to those of the dimension under reduction with respect to a subgroup and under reduction of a direct product. The second order index is shown to be Dynkin’s index, multiplied by the rank of the group. Explicit formulas are derived for the fourth order index. A few reduction problems are solved with the help of higher indices as an illustration of their utility.

Minimal degree of primitive permutation groups

Abstract: If G is a permutation group on a set Ω of n points then the minimal number c of points of Ω permuted by non-identity elements of G is called the minimal degree of G. If G is primitive then Jordan (1871) showed that as n gets large so does c. Later in 1892 and 1897, Bochert obtained a simple bound for c in terms of n provided that G is 2-transitive and is not the alternating or symmetric group: he showed that c ⩾ n/4−1 (in 1892), and c ⩾ n/3−2√n/3 (in 1897). This paper is the result of our efforts to obtain simpler bounds than those of Jordan when G is primitive but not 2-transitive. We show that if G is primitive on Ω of rank r ⩾ 3 and minimal degree c, and if nmin is the minimal length of the orbits of Gα in Ω-{α}, where α e Ω, then c⩾nmin/4+r−1. Moreover as two corollaries of the result we show that if either G has rank 3, or if G is 3/2-transitive then c is of the order of √n, where n=|Ω|, which is better than the bounds of Jordan.

The largest degrees of irreducible characters of the symmetric group

Abstract: The largest irreducible degrees and the partitions associated with them are tabulated for the symmetric group En for n up to 75. Analytic upper and lower bounds are derived for the largest degree. Introduction. A question has been raised by Bivins and others [2] -namely: For what irreducible representations of the symmetric group En does the degree attain its maximal value and how does this maximum behave for large n? This was apparently motivated by the practical considerations of number overflow in the computer but the same question arises in connection with sorting [1]. Each irreducible representation is associated with a partition a = (a1, a2, . .. ak), al > a2 > * * * > ak > O, of n. (We shall use a E n to mean that a is one of the Pn partitions of n.) Its degree is given by [6, p. 61]: da= n! [11i n2 > * * > nk, where n, is the current number of votes for candidate i (i = 1, . . ., k) with finally n, = a (i = I1, . . .,~ k). Computation of the Maximal Degree. The calculations were made at Edinburgh University on a 4K 12-bit word length PDP8 computer using a multi-length routine for expansile integer multiplication. The strategy is straightforward. For increasing n, partitions of n are generated in natural order (n first and In last) as described in [11]. If a partition, a, precedes or coincides with its conjugate then the degree da is computed as in the procedure degree of [9] but exponent arithmetic is used retaining integers throughout and avoiding unnecessary overflow. A description of exponent arithmetic appears in [10] but this description is slightly different from that used in this application, and the algorithm given there is a little garbled. Three arrays are declared, ex, hfac, Ifac [2: N] , where N is the largest integer occurring as a natural factor (here N is at most 75); ex [n] contains the exponent of n in the result and for all n < N, hfac [n] contains the largest prime factor of n and Ifac[n] contains the other factor. After initialization, the expression is evaluated by modifying the exponents in ex. For example, to divide by k!: for i := 2 step I until k do ex [i] := ex [i]-1; Received June 5, 1975; revised September 30, 1975. AMS (MOS) subject classifications (1970). Primary 20-04, 20C30; Secondary 05A15.

Normal transformation semigroups

Abstract: Many transformation semigroups S over a set X have the inner-automorphism property: their automorphisms are precisely those mappings of the form g-1 · g: α → g-1αg(α∈S) where g is a fixed permutation of S. For example, the full transformation semigroup, any two sided ideal of this semigroup, and, with some exceptions for small cardinals, the alternating group and the symmetric group itself. See Malcev (1952) and Scott (1965, Chapter 11). In this paper we determine all transformation semigroups over finite X with this property. In fact, we do more: we characterize all transformation semigroups invariant under the mappings g-1 · g. We refer to these as -normal transformation semigroups.

On the number of permutations of special form

Abstract: It is shown that the number of permutations for which the equation , where ( is the symmetric group of degree ) and is a fixed natural number, has at least one solution is asymptotically equal to as where is a constant depending only on , and is the Euler function.Bibliography: 4 titles.

Efficient CI calculations utilizing the symmetric group

Abstract: Gallup has recently given a solution to the problem of determining an orthogonal representation matrix for an arbitrary permutation of the symmetric group in terms of sandwich representations. In this paper it is shown how this provides a means for the precalculation and storage of the necessary distinct coefficients needed in the construction of the various Hamiltonian matrix elements of an orthogonal orbital CI calculation. This method has the advantage over those published previously in that the spin degeneracy problem is handled more efficiently. It is shown that this procedure can be utilized with both the ’’construction and storage’’ algorithms and, also, the recently developed ’’direct vector’’ methods when dealing with large Hamiltonian matrices.

A group theoretical approach to geminal product matrix elements

Abstract: Expressions enabling systematic compilation of Hamiltonian and overlap matrix elements for an antisymmetrized multiterm geminal product trial function are derived, using double coset (DC) decompositions and subgroup adapted irreducible representations of the symmetric group, SN. The trial function may describe an even electron atomic or molecular system in any total spin eigenstate, and the geminals may be nonorthogonal, have arbitrary permutational symmetry, and be explicit functions of interelectronic distance. A DC decomposition is used to factor out permutations not exchanging particle labels between geminals (elements of the interior pair group, Sn2, n≡N/2). This reduces the sum over N! permutations to a sum over DC generators. If the irreducible representation λ (S) of SN is adapted to Sn2 each geminal is projected into its singlet or triplet component. The DC generators are chosen such that each has the form QP, where Q permutes odd particle labels only and P is a permutation of geminals (element o...

The α-regular classes of the generalized symmetric group

Abstract: The α-regular classes of any finite group G are important since they are those classes on which the projective characters of G with factor set α take non-zero value, and thus a knowledge of the α-regular classes gives the number of irreducible projective representations of G with factor set α (see [4]). Here we look at the particular case of the generalized symmetric group Cm wr Sl. The analogous problem of constructing the irreducible projective representations of Cm wr Sl has been dealt with in [6] by generalizing Clifford's theory of inducing from normal subgroups, but unfortunately, it is not in general possible to determine the irreducible projective characters (and hence the α-regular classes) by this method.

The group algebra of the infinite symmetric group

Abstract: The rational group algebra of the infinite symmetric group is studied using Young diagrams. Maximal and prime ideals are characterized and the maximal condition on ideals is proved.

Specht modules and the radical of the group ring over the symmetric group γp

Abstract: (1976). Specht modules and the radical of the group ring over the symmetric group γp. Communications in Algebra: Vol. 4, No. 5, pp. 435-457.

Introduction to group theory

Abstract: I Definition of a Group and Examples.- 1 The abstract group and the notion of group isomorphism.- a. Sets and mappings.- b. Algebraic systems.- c. Semigroups.- d. Groups.- e. Isomorphism.- f. Cyclic groups 7.- Examples and exercises.- 2 Groups of mappings. Permutations. Cayley's theorem.- a. Composition of mappings.- b. Permutations.- c. Cycles.- d. Transpositions.- e. Subgroups.- f. Cayley's theorem. The group table.- Examples and exercises.- 3 Arithmetical groups.- a. Facts of number theory.- b. Residue classes modulo m. Euler's function.- c. The unit group of a ring.- d. Matrix residue class groups (mod m).- e. The case m = 2.- Examples and exercises.- 4 Geometrical Groups.- a. Rotations and reflexions.- b. The dihedral groups.- c. Rotations and reflexions in space.- d. The polyhedral groups.- e. The groups of the octahedron and of the tetrahedron.- Examples and exercises.- II Subsets, Subgroups, Homomorphisms.- 1 The algebra of subsets in a group.- a.-c. Product and inverse of subsets.- d. Subgroup generated by a subset.- e. Two disjoint subsets covering a group.- Examples and exercises: Frattini subgroup.- 2 A subgroup and its cosets. Lagrange's theorem.- a. Cosets. Lagrange's theorem.- b. Index theorems.- c. Poincare's theorem.- d. Finite cyclic groups.- Examples and exercises: Multiplicative group of a finite field. The icosahedral group. Frattini subgroup. Abelian groups.- 3 Homomorphisms, normal subgroups and factor groups.- a. Homomorphism, epimorphism, monomorphism.- b. Kernel.- c. Natural homomorphism and factor group.- d. Canonical product.- Examples and exercises.- 4 Transformation. Conjugate elements. Invariant subsets.- a. Conjugacy.- b. Invariance..- c. The classes.- d. The normalizer.- e. Generalization of Cayley's theorem.- Examples and exercises: Transformation in permutation groups. Geometry in a group.- 5 Correspondence theorems. Direct products.- a.-b. Theorems.- c. The internal direct product.- d. Generalization to more than two factors.- e. The external direct product.- f. The restricted direct product.- Examples and exercises.- 6 Double cosets and double transversals.- a. A counting formula.- b. Double cosets.- c. Double transversals.- d. A group of double cosets.- Examples and exercises.- III Automorphisms and Endomorphisms.- 1 Groups of automorphisms. Characteristic subgroups.- a. Generalities.- b. Inner automorphisms.- c. Characteristic subgroups.- d. Characteristically simple groups.- e. ?-invariance.- Examples and exercises: Automorphism groups of some special groups. Simplicity of the alternating groups. The kernel or nucleus of a group. The Frattini subgroup, a characteristic subgroup.- 2 The holomorph of a finite group. Complete groups.- a. Definition of the holomorph.- b.-c. The holomorph as a permutation group.- d. Is the holomorph minimal?.- e. Complete groups.- Examples and exercises: Completeness of the symmetric groups. Equicentralizer systems in groups.- 3 Group extensions.- a. The semi-direct product.- b. The external semi-direct product.- c. Are there other solutions to the extension problem ?.- d.-e. Construction of a normal extension.- Examples and exercises: Complement. Splitting extension.- 4 A problem of Burnside: Groups with outer automorphisms leaving the classes invariant.- a. Preliminaries on the groups L2n.- b. The groups L4n and L8n.- c. Automorphism associated with a subgroup of index 2.- d. Proof of the theorem concerning L8n.- Examples and exercises.- 5 Endomorphisms and operators.- a. Endomorphisms.- b. The endomorphism ring of an abelian group.- c. Operator domain of a group. Fully invariant subgroups.- Examples and exercises.- IV Finite Series of Subgroups.- 1 The fundamental concepts of lattice theory.- a. Partially ordered sets.- b. Lattices.- c. Partially ordered set and lattice.- d. Modular lattices and distributive lattices.- Examples and exercises.- 2 Lattices of subgroups.- a. The lattice of all subgroups of a group.- b. The lattice of admissible subgroups.- c. The lattice of normal subgroups.- d. The Lemma of Zassenhaus.- Examples and exercises.- 3 The theory of O. Schreier.- a. Chains and series of subgroups.- b. Refinement of series. Schreier's theorem.- c. The theorem of Jordan and Holder.- d. Applications.- e. Solvable groups.- Examples and exercises: Maximum normal subgroups.- 4 Central chains and series.- a. The ascending central chain.- b. The upper central chain.- c. Nilpotent groups.- d. Mixed commutator subgroups.- e. The lower central chain.- Examples and exercises.- V Finite Groups and Prime Numbers.- 1 Permutation groups.- a. Action of a group on a set.- b. Transitivity.- c. Stabilizers.- d. Application.- e. Multiple transitivity. Imprimitivity.- f. Sylow's first theorem.- Examples and exercises: Bertrand's theorem.- 2 Sylow's theorems.- a. Cauchy's theorem.- b. The class equation.- c. Theorem 1.- d. Theorem 2.- e. Theorem 3.- f. Theorem 4.- Examples and exercises: Landau's theorem. Theorems on finite p-groups. The groups of order pq.- 3 Finite nilpotent groups.- a. A direct product of p-groups.- b. Necessity.- c. Maximal subgroups. Schmidt's theorem.- d. The Frattini subgroup.- Examples and exercises: Gaschutz's theorem.- 4 The structure of finite abelian groups.- a. Existence of a basis.- b. Uniqueness (invariance) of the orders of the basis elements.- c. Application to the construction of abelian groups.- Examples and exercises: The residue class group ? m.- Appendix Hints or Solutions to Some of the Exercise Problems.- Author index.

On a generalization of frobenius' theorem to infinite groups

Abstract: In this paper the following theorem is proved. Theorem. Suppose is a group, is a subgroup, and is an element of prime order in such thata) is a Frobenius pair, i.e. for all ;b) for any the group is finite. Then , where is a periodic group containing no -elements, and either possesses a unique involution or .Examples of periodic groups are given to show that the conditions and b) are essential restrictions in the theorem.It is proved that in the class of periodic biprimitively finite groups the existence in a group of a Frobenius pair already implies that and admits a partition, i.e. .Bibliography: 14 titles.

Inseparable finite solvable groups

Abstract: A finite group is called inseparable if the only normal subgroups over which it splits are the group itself and the trivial subgroup. Let E be the formation of finite solvable groups with elementary abelian Sylow subgroups. This note establishes the fact that, up to isomorphism, there is exactly one nonnilpotent inseparable solvable group in which the E-residual is a metacyclic p-group. A finite group is called inseparable if the only normal subgroups over which it splits are the one-element subgroup and the group itself (see [1]). These groups are of interest since an arbitrary finite group G, I G = # 1, is expressible as a product of subgroups, G = AnAn-I ... A1, for which (a) Ai is inseparable and IAI # 1, for i = 1, .. ., n, (b) (An A1+ I1(A n . A1) forj = 1, .. ., n 1, and (c) A ** A1 = [An ... A+J]A forj = 1,..., n 1. Previous results [1] suggest that the splitting of a group G over its normal subgroups seems to be determined by the E-residual, GE, for the formation E of groups having elementary abelian Sylow subgroups. A minimal inseparable nonnilpotent solvable finite group G with GE as a metacyclic p-group is known [1] to be isomorphic to the group in the following example. EXAMPLE. Let H = represent the quaternion group and K = . There exists a homomorphism 9: K -* Aut(H) having Ker(9) = and K9 isomorphic to the symmetric group of degree 3. Form the partial semidirect product G of H by K such that H n K = Ker(9) = Z(H). Then G is inseparable. The purpose of this note is to establish that each nonnilpotent inseparable finite solvable group G for which GE is a metacyclic p-group is isomorphic to the group in the example. Only finite solvable groups are to be considered. The notation may be found in [3] with the exception that G = [A]B denotes the splitting of a group G over a normal subgroup A by a subgroup B. A group is called a reduced product of a normal subgroup A by a subgroup B if G = AB and A C is a proper subgroup of G for each proper subgroup C of B. Presented to the Society, January 23, 1976; received by the editors February 16, 1976. AMS (MOS) subject classifications (1970). Primary 20D10, 20D40.

Erratum to "On the Automorphism Group of a Lie Group"

Abstract: D. Z. Djokovic and G. P. Hochschild have pointed out that the argument in the third paragraph of the proof of Theorem 1 is insufficient. The fallacy lies in assuming that the fixer of S in K(S) is an algebraic subgroup. This is of course correct for complex groups, since a connected complex semisimple Lie group is algebraic, but false for real Lie groups. By Proposition 1, the theorem is correct for real solvable Lie groups and the given proof is correct for complex groups, but the theorem fails for real semisimple groups, as the following example, worked out jointly with Professor Hochschild, shows. Let G = SL(2, R); then Gc = SL(2, C) is the complexification of G and the complex conjugation T in SL(2, C) fixes exactly SL(2, R). Let