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.

scholarly journals Duality for finite abelian hypergroups over splitting fields

1979 ◽  
Vol 20 (1) ◽  
pp. 57-70 ◽  
Author(s):  
J.R. McMullen ◽  
J.F. Price
Keyword(s):  
Finite Group ◽  
Duality Theory ◽  
Abelian Groups ◽  
Conjugacy Classes ◽  
Finite Abelian Groups ◽  
Precise Sense ◽  
Classical Duality ◽  
A Finite Group

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

A First Approximation to {X, Y}

1969 ◽  
Vol 21 ◽  
pp. 702-711 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Homotopy Theory ◽  
Abelian Groups ◽  
Classification Theorem ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
First Approximation ◽  
Image Position

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.Our aim in this paper is to prove the following theorem.THEOREM. Iƒ ℭ is a class of finite abelian groups and(i) πi(Y) ∈ℭ for i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z)∈ ℭ for i > n + k,ThenThis statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.

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.

Explicit homotopy equivalences in dimension two

2002 ◽  
Vol 133 (3) ◽  
pp. 411-430 ◽  
Author(s):  
F. E. A. JOHNSON
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Finitely Generated ◽  
Stable Class ◽  
A Finite Group

Let G be a finite group; by an algebraic 2-complex over G we mean an exact sequence of Z[G]-modules of the form:E = (0 → J → E2 → E1 → E0 → Z → 0)where Er is finitely generated free over Z[G] for 0 [les ] r [les ] 2. The module J is determined up to stability by the fact of appearing in such an exact sequence; we denote its stable class by Ω3(Z); E is said to be minimal when rkZ(J) attains the minimum possible value within Ω3(Z).

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.

scholarly journals Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

2020 ◽  
pp. 1-7
Author(s):  
Omar Tout
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Abelian Groups ◽  
Symmetric Groups ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Finite Abelian Groups ◽  
A Finite Group

Abstract It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

A Spectral Sequence for Cohomotopy

1969 ◽  
Vol 21 ◽  
pp. 712-729 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Abelian Group ◽  
Spectral Sequence ◽  
Prime Number ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Image Position

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.Theorem. Suppose that(i) Hi(X;Z) = 0 for i > k,(ii) for i > k – dThen there exists a spectral sequence withand the differential is given by

scholarly journals COMMUTATIVITY DEGREE OF A CLASS OF FINITE GROUPS AND CONSEQUENCES

2013 ◽  
Vol 88 (3) ◽  
pp. 448-452 ◽  
Author(s):  
RAJAT KANTI NATH
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Finite Abelian Groups ◽  
A Finite Group ◽  
Open Question

AbstractThe commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

On transiso graph

2015 ◽  
Vol 08 (04) ◽  
pp. 1550070 ◽  
Author(s):  
Vipul Kakkar ◽  
Laxmi Kant Mishra
Keyword(s):  
Finite Group ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Dihedral Groups ◽  
A Finite Group

In this paper, we define a new graph [Formula: see text] on a finite group [Formula: see text], where [Formula: see text] is a divisor of [Formula: see text]. The vertices of [Formula: see text] are the subgroups of [Formula: see text] of order [Formula: see text] and two subgroups [Formula: see text] and [Formula: see text] of [Formula: see text] are said to be adjacent if there exists [Formula: see text] [Formula: see text] such that [Formula: see text], where [Formula: see text] [Formula: see text] denote the set of all NRTs of [Formula: see text] in [Formula: see text]. We shall discuss the completeness of [Formula: see text] for various groups like finite abelian groups, dihedral groups and some finite [Formula: see text]-groups.

Sign in / Sign up

Export Citation Format

Share Document