Presentations of the Trefoil Group

1973 ◽  
Vol 16 (4) ◽  
pp. 517-520 ◽  
Author(s):  
M. J. Dunwoody ◽  
A. Pietrowski
Keyword(s):  
Exact Sequence ◽  
Free Group ◽  
Image Position

A presentation of a group G is an exact sequence of groupswhere F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.

Quasi-Splitting Exact Sequence

1971 ◽  
Vol 23 (3) ◽  
pp. 503-506
Author(s):  
Hsiang-Dah Hou
Keyword(s):  
Exact Sequence ◽  
Full Subcategory ◽  
Abelian Category ◽  
Image Position

Let R be a ring with 1 ≠ 0 and α, β, γ R-endomorphisms of R-modules A, B, and C respectively. We shall denote by M(R) the category of R-modules, and by End(R) the category of R-endomorphisms. For objects α and β of End(R) a morphism λ: α → β is an R-homomorphism such that λα = β λ. We shall denote by Idm(R) the full subcategory of End(R) whose objects are idempotents. Idm(R) is an abelian category, ker, coker and im are constructed in the naive way and henceis exact in M(R) if and only ifis exact in Idm(R), where the domains of α,β, and γ are A, B, and C respectively. One sees that End (R) as well as Idm(R) is abelian.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

Massey Products and Lower Central Series of Free Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 322-337 ◽  
Author(s):  
Roger Fenn ◽  
Denis Sjerve
Keyword(s):  
Free Group ◽  
Differential Calculus ◽  
Free Groups ◽  
Lower Central Series ◽  
Central Series ◽  
Massey Product ◽  
Massey Products ◽  
One Dimensional ◽  
Image Position

The purpose of this paper is to continue the investigation into the relationships amongst Massey products, lower central series of free groups and the free differential calculus (see [4], [9], [12]). In particular we set forth the notion of a universal Massey product ≪α1, …, αk≫, where the αi are one dimensional cohomology classes. This product is defined with zero indeterminacy, natural and multilinear in its variables.In order to state the results we need some notation. Throughout F will denote the free group on fixed generators x1, …, xn andwill denote the lower central series of F. If I = (i1, …, ik) is a sequence such that 1 ≦ i1, …, ik ≦ n then ∂1 is the iterated Fox derivative and , where is the augmentation. By convention we set ∂1 = identity if I is empty.

scholarly journals Non-abelian exterior products of groups and exact sequences in the homology of groups

1987 ◽  
Vol 29 (1) ◽  
pp. 13-19 ◽  
Author(s):  
G. J. Ellis
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Type Theorem ◽  
Exterior Product ◽  
Products Of Groups ◽  
Hopf Formula ◽  
Definition Of ◽  
Image Position ◽  
Exact Sequences

Various authors have obtained an eight term exact sequence in homologyfrom a short exact sequence of groups,the term V varying from author to author (see [7] and [2]; see also [5] for the simpler case where N is central in G, and [6] for the case where N is central and N ⊂ [G, G]). The most satisfying version of the sequence is obtained by Brown and Loday [2] (full details of [2] are in [3]) as a corollary to their van Kampen type theorem for squares of spaces: they give the term V as the kernel of a map G ∧ N → N from a “non-abelian exterior product” of G and N to the group N (the definition of G ∧ N, first published in [2], is recalled below). The two short exact sequencesandwhere F is free, together with the fact that H2(F) = 0 and H3(F) = 0, imply isomorphisms..The isomorphism (2) is essentially the description of H2(G) proved algebraically in [11]. As noted in [2], the isomorphism (3) is the analogue for H3(G) of the Hopf formula for H2(G).

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

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.

On the relation of connective K-theory to homology

1970 ◽  
Vol 68 (3) ◽  
pp. 637-639 ◽  
Author(s):  
Larry Smith
Keyword(s):  
Exact Sequence ◽  
Natural Transformation ◽  
Homology Theory ◽  
Finite Complex ◽  
Ring Spectrum ◽  
Bott Periodicity ◽  
Ring Spectra ◽  
K Theory ◽  
Image Position ◽  
Natural Way

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectrainducing a morphismof homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequencewhere Z is viewed as a Z[t] module via the augmentationand, is induced by η*in the natural way.

scholarly journals Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields

1954 ◽  
Vol 2 (2) ◽  
pp. 66-76 ◽  
Author(s):  
Iain T. Adamson
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Cohomology Group ◽  
Cohomology Theory ◽  
Normal Subgroups ◽  
Cohomology Groups ◽  
Coset Space ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that ifis an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequencemay be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequenceis exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

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.

The Classification of Algebras by Dominant Dimension

1968 ◽  
Vol 20 ◽  
pp. 398-409 ◽  
Author(s):  
Bruno J. Mueller
Keyword(s):  
Exact Sequence ◽  
Infinite Sequence ◽  
Dominant Dimension ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

Nakayama proposed to classify finite-dimensional algebras R over a field according to how long an exact sequenceof projective and injective R-R-bimodules Xi they allow. He conjectured that if there exists an infinite sequence of this type, then R must be quasi-Frobenius; and he proved this when R is generalized uniserial (17).

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