Showing papers on "Centralizer and normalizer published in 1981"
TL;DR: In this article, it was shown that the centralizer of a finite group can be replaced by an arbitrary element of prime order, and that the involution of the group can then be replaced with a subgroup of class at most two and index bounded in terms of m.
Abstract: In [2] we proved that ifG is a finite group containing an involution whose centralizer has order bounded by some numberm, thenG contains a nilpotent subgroup of class at most two and index bounded in terms ofm. One of the steps in the proof of that result was to show that ifG is soluble, then ¦G/F(G) ¦ is bounded by a function ofm, where F (G) is the Fitting subgroup ofG. We now show that, in this part of the argument, the involution can be replaced by an arbitrary element of prime order.
50 citations
TL;DR: It was proved by Krieger as mentioned in this paper that for an ergodic automorphism T of type III there is a flow (F t ) t ∈ R, whose isomorphism class uniquely determines the weak equivalence class of T.
24 citations
01 Feb 1981
TL;DR: Theorem 3 as discussed by the authors states that every connected Lie subgroup H of G containing T is necessarily closed in G. Theorem 3 together with Borel and Siebenthal classification of connected closed subgroups containing T can be viewed as a partial generalization of [2] to arbitrary compact Lie groups.
Abstract: Introduction. We denote by G a compact Lie group and by T a maximal torus of G. The Lie algebras of Lie groups will be denoted by corresponding German letters. Thus g (resp. t) is the Lie algebra of G (resp. T). The connected closed subgroups H of G such that H D T have been studied and classified by Borel and Siebenthal [1]. It is known that every connected Lie subgroup H of G containing T is closed in G. For this see, for instance, [4, Theorem 8.10.1] or Lemma 2 below. Our main result (Theorem 3) is that every subgroup H of G containing T is necessarily closed in G. This note is motivated by a recent paper of Borevich and Krupeckif [2] where they study the case G = U(n). They take T to be the group of diagonal matrices in U(n) and describe all subgroups H of U(n) containing T. Our Theorem 3 together with Borel and Siebenthal classification of connected closed subgroups containing T can be viewed as a partial generalization of [2] to arbitrary compact Lie groups. By N(H) we denote the normalizer of a subgroup H in G. We set N(T) = N. Similarly, if b is a subalgebra of q then n(b) is its normalizer in g, and N(b) its normalizer in G. If H is a closed subgroup of G then Ho denotes its identity component. If H is any subgroup of G then H denotes its closure.
9 citations
7 citations
TL;DR: For a semilocal ring Λ in which each residue field of the center contains at least seven elements, it has been shown in this paper that the group of diagonal matrices can be represented by a uniquely defined D -net of idealsS, where S is the normalizer of the S -net subgroup.
Abstract: It has been proved (Ref. Zh. Mat., 1978, 9A237) that for a semilocal ring Λ in which each residue field of the center contains at least seven elements we have the following description of subgroups of the full linear group GL(n,Λ) that contain the group of diagonal matrices: for each such subgroup H there is a uniquely definedD -net of idealsS (Ref. Zh. Mat., 1977, 2A288) such that G(S)≤H≤N(S), ,whereN(S) is the normalizer of theS -net subgroup G(S). It is noted that this result is also true under the following weaker assumption: a decomposition of a quotient ring of the ring Λ into a direct sum of full matrix rings over skew fields does not contain skew fields with centers of less than seven elements or the ring of second-order matrices over the field of two elements.
7 citations
01 Jan 1981
TL;DR: In this paper, it was shown that the existence of a regular subgroup is not necessary for many of the arguments, and the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.
Abstract: The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.
6 citations
5 citations
4 citations
TL;DR: In this article, the authors provided an affirmative answer to the question "Do there then exist matrix units { e ij } and a left order D in their centralizer such that R contains all matrices in ∑ j De lj?" and showed that the answer is "no".
3 citations
TL;DR: Theorem 4.4.5 as discussed by the authors is a generalization of Theorem 3.1, which states that a CW-complex does not act freely on an odd-order element of order p, p prime, unless 2(P0)(n + 1), where Es(p) is an integer depending on p. Theorem 5.
Abstract: Restrictions other than rank conditicns on elementary abelian subgroups are found for finite groups acting freely on (Sn)k, with trivial action on homology. It is shown that elements x of order p, p an odd prime, with x in the normalizer of an elementary abelian 2-subgroup E of G, must act trivially on E unless pl(n + 1). It is also shown that if p = 3 or 7, x must act trivially, independent of n. This produces a large family of groups which do not act freely on (Sn)k for any values of n and k. For certain primes p, using the mod two Steenrod algebra, one may show that x acts trivially unless 2(P0)(n + 1), where Es(p) is an integer depending on p. Introduction. In this paper, we will describe some restrictions of the structure of finite groups G which act freely on spaces having the homotopy type k Sn . For the case of k = 1, it is well known that all abelian subgroups of G are cyclic, and this condition is sufficient to allow the construction of a finite CW-complex on which G acts freely. In [2], it was proven that for k arbitrary, the elementary abelian 2-subgroups of G must have rank 1, more subtle restrictions than rank conditions are necessary. The following is known. THEOREM [5]. A4 does not act freely on Sn X Sn, with trivial action on mod 2 homology.2 In this paper we apply the results of [2] to generalize this result considerably; thus, the main theorem, Theorem 4.4. THEOREM. Let G be a semidirect product Z/3 x T (Z/2 x Z/2) or Z/7 x T (Z/2 x Z/2 x Z/2), with nontrivial action. Then G does not act freely on a finite CW-complex X, X Hk= S', with trivial action on mod 2 homology, for any values of k and n. For fixed values of n, we obtain stronger restrictions on G, in Theorem 3.5. THEOREM. Let G act freely on X, X _ ll=k1 Sn, where X is finite, and acts trivially on H*(X; Z/2). Let x E NG(E) be an odd order element of order p, p prime, where NG denotes normalizer and E is an elementary abelian subgroup of G. Then, unless pI(n + 1), x is in the centralizer of E. Moreover, there is a function A: Z -> Z, such that x is in the centralizer unless 2 (d)l(n + 1). (A is nontrival, i.e. A(31) = 3.) Received by the editors September 24, 1979 and, in revised form, March 19, 1980. 1980 Mathematics Subject Classification. Primary 55G10, 57E25. 'Supported in part by NSF Grant MCS-7903192. 2R. Oliver informs me that he can also prove this theorem for (Sn)k. ? 1981 American Mathematical Society 0002-9947/81/0000-0161 /$03.25 449 This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:45:05 UTC All use subject to http://about.jstor.org/terms
3 citations
01 Mar 1981
TL;DR: In this paper, the authors examine commutativity properties of conjugate self-adjoint operators which possess cyclic vectors and show that they are abelian if and only if A = QA*Q, where A* is the adjoint of A. The well-known Hankel operators [4] belong to this class.
Abstract: Let A be a bounded linear transformation on the complex separable Hilbert space H. If there is a conjugation Q on H such that A = QA*Q, we say that A is conjugate selfadjoint. In this note we examine commutativity properties of conjugate selfadjoint operators which possess cyclic vectors. 1. Preliminaries. Let H be a complex Hilbert space with a countably infinite basis, and let (f, g) denote the inner product of two vectors in H. By H ff H we mean the Hilbert space of vectorsf E3 g having inner product (fl D gI,f2 E 92) = (fl,f2) + (g1, g2). A linear manifold is a subset which is closed under vector addition and under multiplication by complex numbers. A subspace is a linear manifold which is closed in the norm topology induced by the inner product. The smallest subspace containing the set U ??of fn) will be denoted by V{fn). If F is a subset of Hilbert space, clos F will denote the closure of F in the norm topology and F' = { gl( g, f) = 0, f E F). Whenever A is a continuous linear transformation on H, its graph F(A) = { fD Af If E H) is a subspace of H ff H. One can easily verify that F(A)= { A*(-f) Ef fIf E H), where A* is the adjoint of A. The germinal idea of representing a linear transformation through its associated graph subspace originated in the work of J. von Neumann [1]. Hereafter we shall refer to a continuous (or bounded) linear transformation as an operator. If A and B are operators on H, we define (A ff B)(f E g) = Af ff Bg. The set of all operators on H that commute with A is called the commutant of A. This algebra will be denoted by (A)'. The double commutant of A, designated {A}", is the algebra of all operators on H that commute with every member of (A }'. It is self evident that (A)' is abelian if and only if (A)' = (A A". A transformation Q on H is said to be a conjugation if Q2 = I and (Qf, Qg) = (g, f) for every f and g in H. Intuitively speaking, Q replaces each element of H by its conjugate with respect to the "real" subspace consisting of all fixed points of Q [3, p. 357]. If there is a conjugation Q such that A = QA*Q, we shall call A conjugate selfadjoint. The well-known Hankel operators [4] belong to this class. Received by the editors July 18, 1980 and, in revised form, January 12, 1981. 1980 Mathematics Subject Classification. Primary 47A05; Secondary 46J99.
TL;DR: In this paper, the decomposition of the tensor product of two principal series representations of G into irreducible tensor products was studied for the simply connected double covering, G = Spin(4, 1), of the DeSitter group.
Abstract: The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, G = Spin(4, 1), of the DeSitter group. The main result is that this decomposition consists of two pieces, T, and Td, where T, is a continuous direct sum with respect to Plancherel measure on G of representations from the principal series only and Td is a discrete sum of representations from the discrete series of G. The multiplicities of representations occurring in T, and Td are all finite. Introduction. Let G = Spin(4, 1) be the simply connected double covering of the DeSitter group, G = KAN an Iwasawa decomposition of G, M the centralizer of A in K, and P = MAN the associated minimal parabolic subgroup of G. For a E M and T E A, a x T is a representation of P via a X T(man) = a(m)T(a) and a representation of the form 7(a, T) = Indp a x T is called a principal series representation of G. The main goal of this paper is to determine the decomposition of the tensor product of two principal series representations of G into irreducibles. It was shown in [7], by using Mackey's tensor product theorem and the Mackey-Anh reciprocity theorem, that this problem reduces to knowing how to decompose the restriction to MA of almost every principal series representation of G and each discrete series representation of G. For a representation v7 belonging to the principal series of G, the restriction of v7 to MA, (7)MAA was determined by using Mackey's subgroup theorem. However, in that paper, we were not able to determine explicitly (7)MA for a representation 'r belonging to the discrete series of G. This we do in ?3 of this paper by using Lie algebraic methods and the realizations of these representations given by Dixmier in [2]. This paper is organized as follows. In ?? 1 and 2 we summarize the main results concerning the structure and representation theory of G that we shall use. In ?3 we determine (7)MA when v7 is a discrete series representation of G. We also include the results of [7] concerning the decomposition of (@)MA when v7 is a principal series representation of G. In ?4 we show how to decompose the tensor product of two principal series representations of G. The main results are contained in Theorem 4. The basic methodology used in this paper to decompose principal series tensor products originates in the works of G. Mackey [6], N. Anh [1], and F. Williams [11]. Received by the editors April 15, 1980. 1980 Mathematics Subject Classification Primary 22E43, 81C40. 'This research was partially supported by a grant from the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0207/$04.75 121 This content downloaded from 157.55.39.118 on Fri, 22 Apr 2016 04:32:41 UTC All use subject to http://about.jstor.org/terms
01 Mar 1981
TL;DR: In this article, the authors simplify the treatment given in the last four sections of [1] by using results of Timmesfeld [7] and Smith [5] on groups with large extraspecial subgroups.
Abstract: In [4], Gorenstein and Lyons reduced part of the classification problem for finite simple groups of characteristic 2 type to the solution of certain standard form problems for odd primes. One such problem was to find simple groups with a standard 3-component of type GL(n, 2), n > 6, subject to a few side conditions. This problem was solved by Finkelstein and Frohardt in [1]. The purpose of this paper is to simplify the treatment given there. In particular, the last four sections of [1] can be replaced by the argument here by using results of Timmesfeld [7] and Smith [5] on groups with large extraspecial subgroups. These results are used elsewhere in the developing characterization of finite simple groups of characteristic 2 type, so the present paper might be incorporated into a shorter overall treatment of such groups. Gorenstein and Lyons use a similar approach in ?8 of [4] to eliminate certain sporadic standard form problems, but their argument is different from ours. To state the main result of this paper, recall the notation of [1]. Let L GL(n, 2), n > 6, and let , r = [n/2], be an elementary abelhan 3-subgroup of L or r such that each b, belongs to a natural GL(2, 2) subgroup of L. The main theorem here is the following.
TL;DR: In this paper, the authors analyzed some commutation properties of the sets of mappings of a vector space X over a division ring K with a conjugation j which are relevant when studying symmetries in quantum mechanics and in elementary-particle physics.