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The properties of the symmetric group are used to deduce matrix elements of one and two-electron operators between molecular states. It is shown that these techniques can be applied to any point group as long as no irreducible representation is more than doubly degenerate.

We exhibit a large class K* of real 2 × 2 matrices of determinant ±1 such that, for nearly all A and B in K*, the group generated by A and B1 (the transpose of B) is the free product of the cyclic groups It is shown that K* contains all matrices of determinant ±1 with integer entries satisfying |b| > |a|, |c|, |d|. This gives a generalization of a theorem of Goldberg and Newman [2]. We also prove related results concerning the dominance of b and the discreteness of the free products .

In [15], O. Taussky-Todd posed the problem of title, namely to findX, Y, ZwhenA, Bare given. Clearly ifX, Y, Zexist thenA, Bare either both invertible or both noninvertible.In section 1, the problem is reviewed in caseA, Bare both invertible. The problem is seen to be fundamentally one of group theory rather than matrix theory. Application of results of Shoda, Thompson, Ree to the general group-theoretical results allows specialization to certain matrix groups.In Section 2, examples and counterexamples are given in caseA, Bare noninvertible. A general necessary condition for solvability (involving ranks) is obtained. This condition may or may not be sufficient. For dimA=2, 3 the problem is settled: there is always a solution in the noninvertible case.

Properties of configuration matrices of group codes for the Gaussian channel are considered. It is shown that the configuration matrix of a code generated by a real-irreducible representation is a scalar multiple of an idempotent matrix. A spectral resolution of the configuration matrix of a code generated by an arbitrary real representation is given, and all of its eigenvalues are determined. A relatively simple criterion to determine when the group code spans the space is derived.

The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.

G is a finite group,o a commutative ring with unity, andoG the group algebra ofG overo. Several results of Nakayama and Rim follow foroG from functorial properties ofT G andE G wheno is noetherian. In particular,E G n(A,B)≈ExtG n (A,B).

Matrices for which the group inverse exists and is nonnegative are studied. Such matrices are characterized in terms of a generalization of monotonicity. In particular, nonnegative matrices with this property are characterized in terms of their nonnegative rank factorizations.

Conditions are established under which a given irreducible group of matrices over a perfect field ? is an irreducible component of some reducible maximal locally nilpotent group of matrices over ?.

The principal result is that if G is any irreducible subgroup of GL(n, C) (C the complex field) such that |tr(7^)| is uniformly bounded for all A in G and some fixed n × n nonzero matrix T, then G is equivalent to a unitary group. Similar results are proved for certain associated representations of G.

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.

In two papers,t each having the title " On the Continuous Group that is defined by any given Group of Finite Order," BURNSIDE establishes certain results of decided interest and importance. among them being the theoremst of FROBENIUS on the irreducible factors of group-determinants. The object of this paper is the development of the theory of analogous groups in any arbitrary field or domiiain of rationality. In particular, when the field is the general Galois field of order pn, we obtain a doubly-infinite system of finite groups corresponding to each given finite group. An exceptional case not treated here is that of a field having a modulus which is a factor of the order of the given finite group. ? BURNSIDE bases his woik upon several theorems proved by means of the LIE theory of continuous groups. The corresponding theorems for an arbitrary field are here derived by simple rational processes (?? 2, 3). The auxiliary theorems on invariant-factors (the i; Elementartheiler" of WEIERSTRASS) are established in ? 4 by mneans of the canonical form of a linear transformation in any field. The later developments (?? 5-7) run parallel to the corresponding parts of BURNSIDE'S treatment, but include essential modifications. The results find application in the problem of the representation of a given finite group as a group of linear transformations in a given field upon the smallest number of variables. That the introduction of the concept of a field gives rise to a generalization of KLEIN'S normnal problem may be illustrated by the fact that a given group may be represented as a modular group upon a smaller number of variables than is possible for a representation as an algebraic linear group.

It is the purpose of this paper to find in terms of parameters the most general matrix X which is K-commutative with respect to a given matrix A. The proofs will yield a method of rational construction for such a matrix X.

ŜU2, a group of unitary operators isomorphic to the 2-dimensional unimodular unitary group, is the dynamical invariance group of the 2-dimensional harmonic oscillator. The corresponding group ring Open image in new window is used to classify completely the eigenfunctions, to represent arbitrary operators having nonvanishing matrix elements only within irreducible representations as unique functions of some special ring elements, and to construct general irreducible tensor operators whose matrix elements obey the known selection rules (Wigner-Eckhart theorem).

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) byand, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

The generators of the group or group algebra are used in an analog of the Lie‐Cartan method, which can be applied to finite or infinite groups. This gives a mean for reduction of an arbitrary group representation, using only the matrix representatives of the generators. It is a set of algorithms using pivotal condensation and can easily be coded as a digital computer program. Connections with Lie‐Cartan theory are suggested, the reduction of the symmetric group discussed, and methods for the reduction of representations of the n × n unitary, orthogonal, and proper orthogonal groups suggested.

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.

The direct object of this paper is to prove that every possible finite group which contains an abelian subgroup of half its own order can be represented by square matrices all of whose elements are equal to zero with the exception of those which appear in one of the diagonals, and these are all ordinary complex numbers. The fact that every finite group which can be represented by such matrices must contain an abelian subgroup of half its own order is at once evident. The proof of the stated theorem may be based upon several interesting theorems relating to a possible choice of the independent generators of an abelian group. As these auxiliary theorems seem to be quite fundamental, we proceed to develop them in a somewhat more general form than would be necessary for the particular application in view.

An algorithm is presented whereby an N‐dimensional orthogonal matrix can be represented in terms of ½N(N − 1) independent parameters θk(ν)[ν=2,3,…,N;k=1,2,…,(ν−1)]. The parameters have the character of angles, whose compact domains are defined in a manner such that there exists a one‐to‐one correspondence between the points in the parameter space and the group of orthogonal matrices. Explicit formulas are given which express all matrix elements in terms of the angles, and formulas are given which express the angles in terms of the matrix elements. Special choices of angles give block‐diagonal matrices. For three‐dimensional matrices, the parametrization is equivalent to that of Euler.

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

We consider here the problem of determining when two elements in a free group will have the same character under all possible representations of the given group in the special linear group of 2 x 2 matrices with determinant 1. Problems involving the representation of free groups in terms of 2 x 2 matrices with real entries and determinant 1 have been investigated by R. Fricke in connection with certain problems in the theory of Riemann surfaces [2]. Fricke’s main concern is to find those representations which define discontinuous groups of the conformal self-mappings of the upper half-plane. His method is based on arguments in non-Euclidean geometry, and has only recently been made fully rigorous and transparent by Linda Keen [4]. Apart from the analytic problems inherent in Fricke’s method there are several algebraic problems which have been answered by Fricke either incompletely or not at all. It is the algebraic aspects of Fricke’s theory which we shall investigate here. Independently of Fricke’s problems however, the subsequent development may be considered as a straightforward contribution to the representation theory of free groups. We shall consider the following general situation. K will denote a fixed commutative ring with identity of characteristic zero, and SL(2, K) the group of all 2 x 2 matrices with determinant 1 and entries from K. It will be shown that the character of any element in a free group under an arbitrary representation of the group in SL(2, K) can be constructively represented as a polynomial expression with integer coefficients in the characters of certain simple products of the group generators. Two polynomials will represent the same character if and only if they are congruent modulo a member of a certain class of ideals. This will reduce the problem of determining whether two elements have the same character to the problem of determining the structure of these ideals. The structure of the ideals will then be determined for the free groups of rank 1, 2, and for the free group of rank 3 under restricted conditions on K. We shall show that if two elements u, v in a free group F have the same character under all possible * The work for this paper was done while the author was at the Courant Institute of Mathematical Sciences and was supported by the National Science Foundation Grant GP-28538. Reproduction in whole or in part is permitted for any purpose of the United States Government. 635

By the rank of a transitive permutation group we mean the number of orbits of the stabilizer of a point thus rank 2 means multiple transitivity. Interest is drawn to the simply transitive groups of "small" rank > 2 by the fact that every known finite simple group admits a representation as a primitive group of rank at most 5 while not all of these groups have doubly transitive representations. In this paper we consider finite transitive groups of rank 3, a class of groups which seems to have received little direct attention.

In this paper we study finite transitive groups G acting on a set Q. The results, which are trivial for multiply-transitive groups, directly generalize parts of the discussion of rank-3 groups in [4] and [5l. There arc close connections with Feit and Higman’s paper [2]. For each a EQ let us choose a G,-orbit d(a) # {CZ} so that d(a)8 = d(a”) for all a E Sz and g E G. Relative to LI we introduce a distunxe in Q based on taking the points of d(u) to be at distance 1 from a (see Section 1). The maximum distance we call the diameter of G. A necessary and sufficient condition for G to be primitive is that the diameter be finite with respect to every d. It is important to note, however, that finiteness of the diameter with respect to a single d does not imply primitivity. We study the matrix M of intersection numbers of A (defined in Section 4). M is irreducible if and only if G has finite diameter with respect to A, and in this case the subdegrees and diameter are determined by M. The minimum polynomial of M is shown to coincide with that of the incidence matrix A of d, and it is shown how to compute the trace of A*, q 3 0, in terms of M. This means that if p(x) is a polynomial such that p(M) = 0 and if 0 is a root of p(x), then the multiplicity of 0 as an eigenvalue of A is determined by M. In case the minimum and characteristic polynomials of M coincide, we show that M has simple eigenvalues, from which it follows that the irreducible constituents of the permutation representation have multiplicity 1 and that the degrees of these constituents are determined by M. In this case there is a one-to-one correspondence between the eigenvalues of M and the irreducible constituents of the permutation representation, which preserves conjugacy. The new simple group of order 750, 560 discovered by Janko [6] provides an example in which two of the constituents are conjugate even though the subdegrces are distinct.

To discuss the properties of a group of finite order some concrete form of representation of the group is necessary, except perhaps in the simplest cases. What are called the abstract defining relations (viz., a system of relations of the form A i = 1, B j = 1, . . . . . . A p B q . . . = 1, A p' B q' . . . = 1, . . . . . between a system of non-commutative symbols A, B, . . . , which are necessary and sufficient to ensure that only a finite number of distinct products can be formed from them) no doubt contain implicitly in the most concise form all the properties of the group. To establish the properties, however, on this basis is not in general practicable. For every group there are an infinite variety of possible concrete representations; and in general for an adequate discussion of the properties of the group several of them have to be made use of. In a limited class of cases, including, however, several groups of great importance in analysis, a representation as a group of space-collineations is available. In all such cases it would be expected that this form of representation, as affording scope for space-intuition, would certainly be one of those chosen for discussion. Except, however, as regards the so-called groups of the regular solids, i. e. , groups of rotations round a point, this has not been done. It is proposed in this memoir to discuss the simple group of order 25920 entirely from the point of view of projective geometry. The existence of a group of collineations of this order is not assumed, but is shown to follow from the existence of a remarkable configuration of points, lines and planes in space. This configuration itself arises naturally in connection with a much less complex group of space-collineations. The method followed throughout is synthetical and constructive. To avoid unduly burdening the earlier part of the memoir, it is assumed that the projective groups of finite order on the straight line have been established (as they can be) without appeal to analysis. Further, the simpler properties of the permutation-groups of 4, 5 and 6 symbols are taken as known.

The representation theory of the group generated by the Dirac matrices is studied. It is shown that the Fierz transformation can be expressed in terms of Racah coefficients of this group. A number of generalized Fierz transformations have been found. Simple rules are given for calculating Fierz invariants and anti-invariants.

It is proven that if K is a commutative ring of characteristic pm while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G′ is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.

It is proved that every metabelian torsion-free group with finitely many generators admits a faithful matrix representation over a field of characteristic zero. Bibliography of 5 entries.

The commutativity of certain Hecke algebras of subgroups of PL(2, F) containing the commutator subgroup is established (here F is a discrete normed field with finite field of residues). Application of these results to representations of p-adic groups by automorphic forms on the upper complex halfplane allows one to distinguish subrepresentations belonging to the discrete series. Bibliography: 8 items.

A symmetric expression of the matrix elements of the permutation group defined by Kildyushov is derived. A three terms recurrence relation is also given.