Showing papers on "Symmetric group published in 1969"

The homology of symmetric products

Abstract: In this paper we compute the homology groups for the various symmetric products of any space X of finite type. Thus we complete the calculations begun by M. Morse, Smith and Richardson in the 1930's and carried dramatically forward by N. Nakaoka in a series of papers dating from 1955. Our methods are essentially geometric in nature and are based on a close examination of the geometry of the topological bar construction introduced in [10]. Indeed it was the study of the symmetric products which led to [10], but the exposition given here is selfcontained. (1) The w-fold symmetric product SPm(X) is the set of all unordered «j-tuples of points in X. Equivalently, SPm(X) is the orbit space of the Cartesian product Xm under the action of ?^m, the symmetric group on m letters. It has the quotient topology. Let a base point *elbe given, then there is an inclusion j:SPm(X)<= SPm + 1(X)

K1-theory and the congruence problem

Abstract: The following results are presented: a) A K1-functor of a noncommutative ring with unity is a factor of a general linear group with respect to the subgroup of elementary matrices; b) a description is given of all the subgroups of finite index in a special linear group over the order in a field.

Twisted group algebras. II

Abstract: A generalization of the group algebra of a locally compact group is studied, by expressing the group algebra of a central group extension as the direct sum of closed *-ideals each one of which is isomorphic to such a “twisted” group algebra. In particular, the representation theory of such algebras is associated with the theory of projective representations by studying the representations of the group algebra of the group extension and the associated unitary representations of the group extension.

The symmetry of molecular polarization tensors

Abstract: Polarization tensors are discussed in terms of their intrinsic symmetry group which is a direct product of the point group and the subgroup of the permutation group relevant to the experiment. The study of these latter groups is simplified by use of the isomorphism with certain point groups and permutations of suffixes can be visualized by rotations and reflections of the vertices of various objects in space. The approach unites the previous treatments and provides a means of constructing the bases for the irreducible tensor components. The difficulties introduced by Laplace's equation are explained and the information obtainable from induced birefringence experiments (Kerr and Cotton–Mouton effects) discussed for various systems.

Extensions of group representations over nonalgebraically closed fields

Abstract: Let G be a finite group with HAG a Hall subgroup, let F be any field of characteristic 0 and let E be the character of an irreducible F-representation Se of H. Suppose E is invariant under the natural action of G on the characters of H. As is well known, if Se is absolute irreducible, the character E can be extended to G and in fact in this case it is not hard to see that Se can be extended to an F-representation of G. (For instance, this follows immediately from the results of ?2.) This paper is an attempt to prove (at least when H is solvable) that Se can always be extended to G. We do not succeed in this attempt. We obtain, however, some purely group theoretic conditions which are sufficient to guarantee the extendibility of Se without any assumptions on F. In particular we prove

Character Analysis of U(N) and SU(N)

Abstract: A symmetric group analysis of the characters of U(N) and SU(N) representations yields formulas for (i) the multiplicities of weights in irreducible and tensor product representations, (ii) the coefficients occurring in the Clebsch‐Gordan series decomposition of Kronecker products with an arbitrary number of factors, (iii) the content of irreducible and tensor product representations of U(Σi Ni) with respect to representations of its direct product subgroup, U(N1)⊗U(N2)⊗…≡⊗i U(Ni), and (iv) the content of irreducible representations of U(NM) with respect to irreducible representations of U(N)⊗U(M). In particular, we exhibit formulas for (i), (ii), and (iii) containing only irreducible characters and Frobenius compound characters of the symmetric group. Under the application of an operator of the subgroup, ⊗i U(Ni) with Σi Ni < N, a vector in a representation of U(N) transforms as a linear combination of vectors in irreducible representations of the subgroup. We give formulas for determining the vectors occ...

Applications of s-functional analysis to continuous groups in physics

Abstract: S-functions, as developed by Littlewood, are reviewed with the aim of simplifying the algebra of continuous groups. S-function division is defined and the theory developed to a stage where a computer programme has been written that performs Kronecker products, branching rules, plethysms on two variables, and inner products of the symmetric group.

On the second natural representation of the symmetric groups

Abstract: In [1], the natural representation module of the symmetric groups, hereafter called the first natural representation module of the symmetric groups, was analysed. It is the purpose of this paper to analyse the second natural representation module of the symmetric groups.

Unitary Representations of Generalized Symmetric Groups

Abstract: In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute with Obviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) and then {c 1, c 2, …, cn } generate a normal subgroup Q(n) of order mn and {s 1, s 2, …, s n…1} generate a subgroup S(n) isomorphic to S(n, 1).

Extensions of group representations over fields of prime characteristic

Abstract: Let G be a finite group having a normal Hall subgroup H, let K be a field, and let T be an irreducible (linear) K-representation of H of degree deg T whose character is invariant under the action of G. We say that T is extendible to G if there exists a K-representation S of G such that S(h) = T(h) for all hCH. In [5, Theorem 6] Gallagher proved that T is extendible if K is the field of complex numbers. The case when K is an arbitrary field of characteristic zero is treated by Isaacs in [7]. In this note we show that the arguments in Isaacs' paper can be extended to yield the following result:

An extension of group theory

Abstract: This chapter highlights a mathematical structure that originates from the theory of S -rings, which is based on the group structure and which has been invented to provide group theory with new, yet elementary, tools for investigating the structure of groups. S -rings are certain subrings of the group rings of finite groups. They were discovered by I. Schur and have been used mainly in the theory of finite permutation groups to prove remarkable results in an elementary fashion. Several years of work on S -rings and on a generalized character theory on finite groups have led to a new view of the theory of S -rings. The very simple idea is to look at S -rings not as a special type of rings but as a mathematical structure in its own right, that is, to produce a notion of S -ring homomorphism so as to obtain a category. This concept and the theory arising from it can easily be generalized to arbitrary groups.

Groups defined by permutations of a single word

Abstract: is in fact the identity group. The question arises as to what relations in a presentation for a group force the group to "collapse" to the identity in general. This problem has been shown to be recursively unsolvable by Rabin [2], but in restricting our attention to a very special class of finitely presented groups generalizing the example above an answer is obtained. DEFINITION. Let W= W(xi, ***, x.) be a word in the symbols xl, * * *, xn (not all necessarily explicitly appearing). For uESn, the symmetric group on 1, 2, . . . , n}, uW denotes the word obtained from W by permuting the indices of the xi according to o: xi-*Xu(i). Furthermore for any subset WCSn, we define the group

Semi-isomorphisms of certain infinite permutation groups

Abstract: Let X and Y be infinite cardinal numbers, S(X) the full symmetric group on a set of cardinal X, A (X) the alternating group of finite even permutations on the same set, and S(X, Y) the subgroup of S(X) of all permutations moving fewer than Y elements. A semi-automorphism of a group G is a permutation T of G such that (xyx)T= (xT)(yT)(xT) for all x, yEG. Semi-isomorphism is defined similarly. Dinkines [1] and Herstein and Ruchte [2] showed that any semi-automorphism of S(X, Y) or A (X) was either the restriction T of an inner automorphism of S(X), or was of the form T(-I) where x(-I) =x-1 for all x. Theorem 11.4.6 of [3] states that every automorphism of any group G such that A (X)CGCS(X) is the restriction of an inner automorphism of S(X). In the present paper, we prove the common generalization of these two theorems whose statement is obvious. (See the corollary at the end.)

Some problems in the theory of finite insoluble groups

Abstract: In this thesis, a study is made of finite groups which satisfy the following hypothesis:- (*) G is a finite group admitting an automorphism α of order r with a fixed point subgroup of order q, where q and r are distinct prime numbers. The main result is Theorem A . Let G be a group satisfying hypothesis (*). Assume either that q is odd or that q = 2 and r is not a Fermat prime greater than 3. Then G is soluble. First a structure theorem for soluble groups satisfying hypothesis (*) is obtained (theorem B). Solubility of groups satisfying hypothesis (*) is shown under the additional assumption that the symmetric group S 4 is not involved: this serves to prove solubility in general for r = 2 or r =3, and to point to the initial reductions in the general case. The proof of theorem A is by induction on the order of groups satisfying the hypothesis for a given pair (q,r). A minimal counterexample is simple, and the remainder of the proof is to show non-existence. After initial reductions the cases q odd and q = 2 are considered separately. For the case q odd, we require the following theorem which is of independent interest: it is a generalisation of a result of Glauberman, theorem A of ( 12 ). Let p be a prime and P a p-group. Let d(P) denote the largest of the orders of the abelian subgroups of P, and let J(P) be the subgroup of P generated by the abelian subgroups of order d(P). Also, let Qd(p) be the natural semi-direct product of z p × z p , regarded as a vector space, by SL(2,p). We obtain Theorem 9.3 . Let G be a finite group, p a prime, P a Sylow p-subgroup of G, and Q a subgroup of Z(P). If Q is normal in N G (J(P)) and if either (i) p is odd and (p-1) does not divide the index [N(Q):C(Q)] or (ii) Qd(p) is not involved in G, then Q is weakly closed in P with respect to G. For the case q = 2, the arguments involved in the proof of theorem A are mainly character-theoretic: we also obtain information about the exceptional cases. Except as mentioned above, the arguments are entirely group-theoretic.