Showing papers on "Symmetric group published in 1971"

Generators for Alternating and Symmetric Groups

Abstract: Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newman's interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF(2,p) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

Hook representations of the symmetric groups

Abstract: In this paper we are concerned with the representation theory of the symmetric groupsover a field K of characteristic p . Every field is a splitting field for the symmetric groups. Consequently, in order to study the modular representation theory of these groups, it is sufficient to work over the prime fields. However, we take K to be an arbitrary field of characteristic p , since the presentation of the results is not affected by this choice. S n denotes the group of permutations of { x 1 , …, x n ], where x 1 ,…, x n are independent indeterminates over K . The group algebra of S n with coefficients in K is denoted by Ф n .

Primitive permutation groups with small orbitals

Abstract: Ga- A 4 or for which G A -~ $4 and is faithful. We also consider primitive groups G which have an orbital A of prime length p > 5 such that IG~l<2p, and reduce the classification problem to that of classifying simple groups with maximal dihedral subgroups of order 2p. The notation and terminology for permutation groups will be that of Wielandt [-8]. We also adopt the notation N(c~)= G~,~(~) and N'(~)= G~uA,(c~) , where A' is the orbital of G paired with A. The definition of orbital is given by D. G. Higman in [2]. 2. In this section we prove a result which, under rather restrictive conditions, gives a strong bound on IG=l. The approach was first pointed out by G. Glauberman for the particular case IAl= 5 and [GA[= 20.

On the solvability of unit groups of group algebras

Abstract: Let FG be the group algebra of a finite group G over a field F of characteristic/? a 0; and let C be the group of units oí FG. We prove that t/is solvable if and only if (i) every absolutely irreducible representation of G at characteristic p is of degree one or two and (ii) if any such representation is of degree two, then it is definable in Fand F=GF(2) or GF(3). This result is translated into intrinsic grouptheoretic and field-theoretic conditions on G and F, respectively. Namely, if Op(G) is the maximum normal ^-subgroup of G and L = G/Op(G), then (i) L is abelian, or (ii) F=GF(3) and L is a 2-group with exactly (\L\ — [£:L'])/4 normal subgroups of index 8 that do not contain V, or (iii) F=GF(2) and L is the extension of an elementary abelian 3-group by an automorphism which inverts every element. Conditions are found for the nilpotency, supersolvability, and ^-solvability of U. This paper presents necessary and sufficient conditions for the solvability of the group U(FG) of two-sided units in the group algebra FG of a finite group G over a field F. The results are essentially those of the author's doctoral dissertation, presented in a more general form. The author is indebted to D. B. Coleman for many helpful suggestions and for continued encouragement. In §1 the contribution of a nilpotent ideal of a ring with 1 to its unit group is considered. Three general results are given in §2 for unit groups of finite-dimensional algebras that are separable modulo their radicals. In §3 there is given a necessary and sufficient condition for the solvability of the unit group of such an algebra. In §4 the group U(FG) is considered. Its solvability is related to the representations of G and the nature of G and F. The groups Op(G) and G/Op(G) are considered here. In §5 the solvability, nilpotency, and /?-solvability of U(FG) are related to the internal structure of G. 1. Let R be a ring with 1 and let the set of two-sided units of R be designated U(R). Theorem 1. Let N be a nilpotent ideal of R. Then the set 1 4TV is a nilpotent normal subgroup of U(R) of class c^a, the index of nilpotency of N. Furthermore, ifR is an algebra over afield F of prime characteristic p, then 1 + N is a p-group with an exponent. Received by the editors September 14, 1970. AM S 1969 subject classifications. Primary 2080, 1640; Secondary 2040.

Maximal abelian subgroups of the symmetric groups

Abstract: Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires. Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdos and Turan deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turan who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn , what kind of properties hold for almost all subgroups of the class?

Spin‐Free Computation of Matrix Elements. I. Group‐Theoretical Computation of Pauling Numbers

Abstract: Spin‐free wavefunctions may be symmetry adapted to a given irreducible representation of the symmetric group SN by a number of different operators. When matrix elements over these symmetry‐adapted wavefunctions are computed, one obtains terms which are group theoretical in nature and terms which are dynamical. In this paper we present formulas for computing the group‐theoretical coefficients for four types of projectors–spin‐free equivalent of the Lowdin projection operator, ξY; structure projector (spin‐free equivalent of valence‐bond‐type functions), κY; sequence‐adapted spin‐free equivalent of the Lowdin projection operator, ξYG; and sequence‐adapted structure projector, κYG. We also give a formula relating the Pauling numbers for the different symmetry projectors to a reference projector.

Codes with simple automorphism groups, II

Abstract: For primesl≧ 11 of the form 2m+1,m an odd prime, the automorphism group of the extended (l+1, (l+1)/2) quadratic residue code overGF(q) is simple. If the automorphism group properly containsPSL 2 (l), then for all primes of the above form the stability group of a point is simple. Application is made to the casesl=7, 11 and 23.

Spin‐Free Computation of Matrix Elements. II. Simplifications Due to Invariance Groups

Abstract: The spin‐free equivalents of the antisymmetrized Lowdin projector, the valence‐bond structure projector, and a number of other symmetry projectors are shown to be invariant to within a phase under the operation of elements of certain subgroups of the spin‐free symmetric group, SN, associated with a spin‐free N‐electron system. These so‐called invariance groups are used to demonstrate equalities among the Pauling numbers thereby reducing the number of Pauling numbers which must be computed. In addition the subgroup of SN whose elements leave a basis ket invariant to within a phase factor are described and used to eliminate redundancy of molecular integrals in the expression for expectation values. By the use of invariance groups of both the projectors and kets, matrix elements over spin‐free operators are simplified by eliminating redundancy with respect to both the Pauling numbers and the molecular integrals.

On the irreducible characters of the Weyl groups

Abstract: In this thesis we study the irreducible characters of the Weyl groups of the simple Lie algebras, in order to give a unified approach to this problem. Chapter one sets up notation. In chapter two we give some known results on the character theory of Weyl groups of type A (the symmetric group) using Weyl subgroups. These are a common feature of Weyl groups and allow us, in chapter three, to generalize to type C. Chapter four deals with type D Which presents a more difficult problem; chapter five is a brief study of the Weyl groups of type B, and finally, chapter six deals with the calculations in the exceptional types G 2 , F4 and E6.

Modular spin representations of the symmetric group

Abstract: 1. Let Γn be the representation group or spin group (4, 9) of Sn. Then the irreducible representations of Γn are of two distinct types. These are (a) ordinary representations, which are the irreducible representations of the symmetric group and (b) spin or projective representations. Corresponding to every partition (λ) = (λ1, λ2, …, λm) of n with λ1 > λ2 > … > λm > 0 there is an irreducible spin representation 〈λ〉 of Γn.

Rank 3 permutation groups with a regular normal subgroup

Abstract: A (p ,n) group G is a permutation group (on a set Ω) which possesses a regular normal elementary abelian subgroup of order pn. The set Ω may be identified with a vector space V on which Go, the stabilizer of a point in G, acts as a subgroup of the general linear group GL(n,p). By a line of a subset ∆ of V, we mean the intersection of ∆ with a one-dimensional subspace of V. The main result (Theorem 1.3.2) concerns (*) - groups, the term we give to rank 3 (p,n; groups in which the stabilizer of a point is doubly-transitive on the lines of a suborbit. The essence or the problem is that of finding those subgroups of PGL (n,p) which have two orbits on the projective space PG (n – 1,p) and act doubly-transitively on one of them. The notion of rank of a permutation group is discussed in 1.1, outline D.G. Higman’s combinatorial treatment of rank 3 groups. Associated with each permutation group having a regular subgroup is a certain S - ring, an algebraic structure which is basic to our theory. In 2.1 we define parameters of a rank 3 S - ring whd.ch coincide with those of any associated rank :3 group. Hence (*) - group with given parameters may be classified by finding all S - rings with the same parameters and then finding the associated (*) - groups. To assist in this task the concepts of residual S-ring and the automorphism group of an S-ring are introduced. Also of great value is Tamaschke’s notion of' the dual S-ring, whi.ch is adapted to use in 2.2. In 3.1 we see how the imposition of conditions of transitivity on a suborbit of a rank 3 (p,n) groups leads to information about the parameters. In 3.3 the various relations connecting the parameters of' a (*)- group are combined to yield specific sets of parameters, all of which are found in §4: to admit rank 3 S - rings. From results concerning the uniqueness of these S – rings, certain finite simple groups are characterised as their automorphism groups, and the proof of the main theorem is completed. A number of results are obtained as by – products in §4:, notably the answer to a question raised by Wielandt and a new representation of the simple group PSL(3,4) as a subgroup of PO-(6,3, leading to an interesting presentation of a recently-discovered balanced block design. §5 is devoted to rank 3 (p,n) groups in which the transitivity condition on Go is replaced by the condition that the associated block design is balanced.

Group‐Theoretic Derivation of Crossing Sum Rules

Abstract: Crossing relations between amplitudes with definite isospin for ππ scattering are expressed in terms of the symmetric group S3. The sum rules, involving s‐channel partial wave amplitudes, are then derived without reference to the Balachandran and Nuyts expansion and are shown to be a direct consequence of the group‐theoretic orthogonality relations. This results in closed expressions for all possible sum rules.

An Exactly Soluble Quantum-Mechanical Three-Body Problem

Abstract: An exact solution is obtained for a model three-body system in which the particles interact through pairwise, harmonic oscillator, and spin-dependent potentials, the solution being achieved by a simple transformation to an appropriate coordinate system. In particular, it is shown that the S-state wave functions of the ground state of the triton span the two-dimensional representation of the symmetric group S3, are orthogonal, and resemble closely the Irving wave forms found so successful in variational calculations.

Groups and automata

Abstract: Four permutation groups are associated with any finite monogenic automaton A , namely the Suschkewitsch group G( A ), the well known automorphism group Aut( A ), the maximal kernel group ┌( A ), and the maximal group homomorphic image Δ( A ) Several results are given relating these groups to the algebraic structure of A and, in the case when A is the reduced acceptor Ac (A*) associated with a prefix code A, to some properties of A Proofs are not included and will be published elsewhere

$p$-solvable groups with few automorphism classes of subgroups of order $p$

Abstract: This paper investigates the relationship between the p-length, 1,(G), of the finite p-solvable group G and the number, a,(G), of orbits in which the subgroups of order p are permuted by the automorphism group of G. If p>2 and ap(G) < 2, it is shown that l,(G);5a,(G). If p=2 and a2(G)=1, it is proved that either lp(G)_?ap(G) or G/02'(G) is a specific group of order 48.

Totally real subfields of $p$-adic fields having the symmetric group as Galois group

Abstract: In this paper, an elementary proof is given of the following proposition: Theorem 1. If Qp is an arbitrary field of p-adic numbers, then it contains normal subfields Ln(2 ≤ n ≤ p) which have symmetric groups Sn as their respective Galois groups over Q, the field of rational numbers. Furthermore, each Ln may be chosen to be totally real.

Chunikhin's existence theorem for subgroups of a finite group

Abstract: We give a simplified proof of a general theorem of Chunikhin on existence of subgroups of a finite group. The proof avoids the technical device of “indexials” which Chunikhin set up for this purpose.