Showing papers on "Symmetric group published in 1968"

Finite groups having only one irreducible representation of degree greater than one

Abstract: In this paper all finite groups having exactly one irreducible K-representation of degree greater than one are determined, where K is an algebraically closed field of characteristic zero. The quaternion group of order eight and the dihedral group of order eight are nilpotent groups with this property, while the symmetric group on three letters and the alternating group on four letters are solvable although not nilpotent examples. The theorem below will show how typical the above examples really are. In the following all groups are finite. If G is a group, let GI denote the derived group of G and Z(G) the center of G. Let K be an algebraically closed field of characteristic zero.

Recoupling coefficients of the symmetric group for shell and cluster model configurations

Abstract: Recoupling coefficients of the symmetric group are defined from the reduction of irreducible representations of this group to direct product representations of particular subgroups. These recoupling coefficients are analysed and shown to be identical to the invariants of unitary groups arising from the reduction of Kronecker product representations. The recoupling coefficients have applications in configurations of several shells or clusters.

The Use of S-Functions in Combinatorial Analysis

Abstract: The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

The Symmetric Group Made Easy

Abstract: Publisher Summary This chapter presents results concerning irreducible representations of the symmetric group, which appears to be unfamiliar to or unappreciated by most chemists and physicists are set forth without proof. These results are closely associated with the familiar methods expounded by Kotani for constructing symmetry-adapted spin functions. Young's diagrams provides explicit rule for obtaining the matrices of the irreducible representations. In order to describe a representation, names are needed for a set of basis vectors. Young tableaux associated with a fixed Young diagram are used to name a set of basis vectors for the corresponding representation. There are two other equivalent sets of symbols that could be used for the same purpose and which are explained, in passing, since they are useful: lattice permutations and Yamanouchi symbols. The topic of lattice permutations is treated by MacMahon and is discussed in the chapter. Yamanouchi symbol is the lattice permutation written backwards.

Weakly closed elements of Sylow subgroups. II

Abstract: The subgroup J(S) was introduced by Thompson in [t0]. Using a slightly different characteristic subgroup of S, Thompson characterized [9] the groups that have normal p-complements for odd primes p. We prove an analogue of his result in Section 7. An element x that satisfies the conclusion of Theorem A is said to be weakly closed in S (with respect to G). We showed in [4] that G is not simple if G is non-cyclic and if a Sylow 2-subgroup S of G contains a non-identity element that is weakly closed in S. We give some applications of Theorem A for p=2 in Section8. Unfortunately, Theorem A cannot be extended to arbitrary finite groups when p=2; the symmetric group of degree four and the simple group of order 168 are counterexamples. Theorem A was originally proved for groups in which the normalizers of non-identity p-subgroups are p-solvable; however, a recent result of Thompson (Theorem 4) serves as a substitute for p-solvability in a minimal counterexample to the general case. We thank Professor Thompson for informing us of his result and for making suggestions which simplified our proofs and led to our results in Theorem A for p = 2. We are grateful to the National Science Foundation for its partial support during the writing of this article.

The Number of Graphs with a given Automorphism Group

Abstract: In this paper, the graphs under consideration may have multiple edges but they do not have loops. We enumerate the number N[H: n, p] of topologically distinct graphs with n vertices and p edges whose automorphism group is the permutation group H. As in (5), this enumeration is considered in the context of the theory of permutation representations of finite groups. We begin with some definitions and notation.

Group algebras with nilpotent unit groups

Abstract: where each Df is a division algebra over F; Af„,.(Z>,) denotes the total matric algebra over £>,-. U(G, F) is thus the direct product of the general linear groups GL(w,-, Di), i=l, • • • , s. Suppose that p t^O. If w, = 1, then the division algebra Z>,is spanned by a homomorphic image of G; hence by [2], Z>, is a field. Thus some »y exceeds 1. For such an n,-, GL(nj, Df) is not nilpotent; hence U is not nilpotent. Suppose that p = 0. If some wtis greater than 1, then as above, U is not nilpotent. Assume that each «,-= 1. Then F(G) is the direct sum of division algebras, so according to [l], G is Hamiltonian and U has the group of nonzero rational quaternions as a subgroup. It is easy to see that this group is not nilpotent; hence neither is U. (b) The following easy result gives a convenient way of handling the sufficiency part of (b) when F is infinite.

The field of multisymmetric functions

Abstract: and denote by k(x)r the field they generate over k. The same symmetric group G now acts on k(x)r by permuting the n sets among each other; that is, g(xia) =x(j), The G-invariant elements are called multisymmetric functions, and again it is a classical result that the field k(x)G they form can be generated by the elementary multisymmetric functions. (These are the coefficients of fl(i+xi U1+ +Xir Ur) when written out as a polynomial in the indeterminates UaX.) But if n, r>1, these elementary functions cannot be independent because there are more than nr of them. Nevertheless,2

Irreducible Representations of the Semidirect‐Product Group Kn = An : Sn and the Harmonic‐Oscillator Shell Model

Abstract: If An denotes the Abelian group of n × n unitary diagonal matrices and Sn the symmetric group represented by n × n permutation matrices, the set of elements ap with a∈An and p∈Sn form a group Kn which is the semidirect product of An and Sn. Irreducible representations of Kn and the chains Kn⊃Sn and Un⊃Kn, with Un being the unitary group in n dimensions, are discussed with applications in the harmonic‐oscillator shell model.

A remark on the group rings of order preserving permutation groups.

Abstract: If G and H are two groups such that their integral group rings Z(G) and Z(H) are isomorphic, does it follow that G and H are isomorphic? This is the isomorphism problem and an affirmative answer is obtained in case G is a sub group of the group of order preserving permutations of a totally ordered set.

On the conversion of the determinant into the permanent

Abstract: Let Mm(F) be the vector space of m - square matrices over a field F. If X belongs to Mm(F), then xij will denote the element occuring in row i and column j of X. Let S m be the symmetric group of degree m and e: S m → F the alternating character on Sm (i.e. e(σ) = 1 or - 1 according as σ is an even or odd permutation).

A Combinatorial Proof of a Conjecture of Goldberg and Moon

Abstract: Let Tn denote a tournament of order n, let G(Tn) denote the automorphism group of Tn, let |G| denote the order of the group G, and let g(n) denote the maximum of |G(Tn)| taken over all tournaments Tn of order n. Goldberg and Moon conjectured [2] that for all n≥1 with equality holding if and only if n is a power of 3. In an addendum to [2] it was pointed out that their conjecture is equivalent to the conjecture that if G is any odd order subgroup of Sn, the symmetric group of degree n, then with equality possible if and only if n is a power of 3. The latter conjecture was proved in [1] by John D. Dixon who made use of the Feit-Thompson theorem in his proof. In this paper we avoid use of the Feit-Thompson result and give a combinatorial proof of the Goldberg-Moon conjecture.

Normal structures and automorphism groups of t-designs

Abstract: Combinatorial configurations known as t-designs are studied. These are pairs ˂B, ∏˃, where each element of B is a k-subset of ∏, and each t-design occurs in exactly λ elements of B, for some fixed integers k and λ. A theory of internal structure of t-designs is developed, and it is shown that any t-design can be decomposed in a natural fashion into a sequence of “simple” subdesigns. The theory is quite similar to the analysis of a group with respect to its normal subgroups, quotient groups, and homomorphisms. The analogous concepts of normal subdesigns, quotient designs, and design homomorphisms are all defined and used. This structure theory is then applied to the class of t-designs whose automorphism groups are transitive on sets of t points. It is shown that if G is a permutation group transitive on sets of t letters and ф is any set of letters, then images of ф under G form a t-design whose parameters may be calculated from the group G. Such groups are discussed, especially for the case t = 2, and the normal structure of such designs is considered. Theorem 2.2.12 gives necessary and sufficient conditions for a t-design to be simple, purely in terms of the automorphism group of the design. Some constructions are given. Finally, 2-designs with k = 3 and λ = 2 are considered in detail. These designs are first considered in general, with examples illustrating some of the configurations which can arise. Then an attempt is made to classify all such designs with an automorphism group transitive on pairs of points. Many cases are eliminated of reduced to combinations of Steiner triple systems. In the remaining cases, the simple designs are determined to consist of one infinite class and one exceptional case.