A non-cyclic one-relator group all of whose finite quotients are cyclic

1969 ◽  
Vol 10 (3-4) ◽  
pp. 497-498 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Normal Subgroup ◽  
Residually Finite ◽  
Defining Relation ◽  
Finite Quotient ◽  
Finite Quotients ◽  
Image Position

Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

Quotient groups and realization of tight Riesz groups

1973 ◽  
Vol 16 (4) ◽  
pp. 416-430 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Normal Subgroup ◽  
Open Interval ◽  
Canonical Homomorphism ◽  
Interval Topology ◽  
Essential Step ◽  
Image Position ◽  
Quotient Groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

scholarly journals Homomorphism theories for universal algebras

1968 ◽  
Vol 16 (1) ◽  
pp. 19-35 ◽  
Author(s):  
Hans-Jürgen Hoehnke
Keyword(s):  
Normal Subgroup ◽  
Unit Element ◽  
Group Rings ◽  
Universal Algebras ◽  
Ring Of Integers ◽  
Image Position

It is well-known that a homomorphism ø(A→B) between groups A and B induces a homomorphism ø*(ZA→ZB) between the corresponding group rings ZA and ZB over the ring of integers Z. The identical congruence O on B and the unit element eB of B can be characterised by the equations x–y = 0 and x–eB = 0 (x,y ∈ B) respectively. Similarly the congruence Γø corresponding to ø and the corresponding normal subgroup of A areand {x∈A1 = A,(x–eA)ø = 0} respectively.

The subnormal structure of general linear groups

1986 ◽  
Vol 99 (3) ◽  
pp. 425-431 ◽  
Author(s):  
L. N. Vaserstein
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Associative Ring ◽  
Inverse Image ◽  
General Linear ◽  
Linear Groups ◽  
Canonical Homomorphism ◽  
General Linear Groups ◽  
Subnormal Structure ◽  
Image Position

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,is the centre of GLnA

scholarly journals Note on a theorem of Magnus

1969 ◽  
Vol 10 (3-4) ◽  
pp. 469-474 ◽  
Author(s):  
Norman Blackburn
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Reduction Theorem ◽  
The South ◽  
Cup Product ◽  
South West ◽  
Image Position ◽  
Independent Parameters

Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.

Dimension subgroups modulo n

1970 ◽  
Vol 68 (3) ◽  
pp. 579-582 ◽  
Author(s):  
Siegfried Moran
Keyword(s):  
Normal Subgroup ◽  
Group Algebra ◽  
Augmentation Ideal ◽  
Arbitrary Group ◽  
Image Position

Let G be an arbitrary group and Zn(G) denote the group algebra of G over the integers modulo n. If δi(G) denotes ith power of the augmentation ideal δ(G) of Zn(G), thenis easily seen to be a normal subgroup of G. It is denoted by Di, n(G) and is called ith dimension subgroup of G modulo n. It can be shown that these dimension subgroups are determined by the dimension subgroups modulo a power of a prime p. Hence we shall restrict our attention to these dimension subgroups.

scholarly journals Projective characters of degree one and the inflation-restriction sequence

1989 ◽  
Vol 46 (2) ◽  
pp. 272-280 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Schur Multiplier ◽  
Original Result ◽  
Image Position ◽  
A Finite Group ◽  
Invariant Element

AbstractLet G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

Finite Quotients of the Automorphism Group of a Free Group

1977 ◽  
Vol 29 (3) ◽  
pp. 541-551 ◽  
Author(s):  
Robert Gilman
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Outer Automorphism ◽  
Permutation Representation ◽  
Finitely Generated ◽  
Outer Automorphism Group ◽  
Finite Quotients

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free group.

A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

1981 ◽  
Vol 33 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Hypergeometric Function ◽  
Jacobi Polynomials ◽  
Defining Relation ◽  
Linearization Problem ◽  
Image Position ◽  
Linearization Coefficients ◽  
Negative Representation

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:(1.1)Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion(1.2)

On Nonabelian H2 for Profinite Groups

1992 ◽  
Vol 44 (2) ◽  
pp. 388-399
Author(s):  
K.-H. Ulbrich
Keyword(s):  
Normal Subgroup ◽  
Exact Sequence ◽  
Profinite Group ◽  
Profinite Groups ◽  
Image Position

Let G be a profinite group. We define an extension (E, J) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of K satisfying: for some open normal subgroup Sof G, and the map is continuous (A being discrete).

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