Characteristic classes for permutation representations

1981 ◽  
Vol 90 (2) ◽  
pp. 265-272 ◽  
Author(s):  
G. B. Segal ◽  
C. T. Stretch
Keyword(s):  
Finite Group ◽  
Characteristic Classes ◽  
Cohomology Groups ◽  
Trivial Representation ◽  
Real Representation ◽  
Finite Dimensional ◽  
Whitney Class ◽  
Image Position ◽  
A Finite Group

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney classin the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and(b) stable, i.e. w(V) = 1 when F is a trivial representation.

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.

Characteristic classes of Galois representations

1990 ◽  
Vol 108 (3) ◽  
pp. 517-522
Author(s):  
A. Kozlowski
Keyword(s):  
Galois Group ◽  
Galois Representations ◽  
Finite Extension ◽  
Principal Result ◽  
Characteristic Classes ◽  
Characteristic Class ◽  
Real Representation ◽  
Whitney Class ◽  
Image Position ◽  
Separable Closure

Let K be a field with char if K≠2 and let Ks denote the separable closure of K and GK the Galois group of the extension Ks/K. If K⊂L is a finite extension and ρ:GL↦Or(R) a (continuous) real representation of GL we have a map ρ:BGL→BO which is used to define Stiefel–Whitney classes wi(ρ) = ρ*(wi). In general if f is any element of H*(BO; ℤ/2) we denote by f(ρ) the characteristic class ρ*(f). Now letbe a genus (see e.g. [9]), for example the total Stiefel–Whitney class w = 1+w1+w2 + … Let K⊂L and ρ be as above and let denote the multiplicative transfer (see e.g. [3, 5, 2, 14, 15]). Our principal result is a generalization of theorem 1 of [3]

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

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.

On Extending Projectives of Finite Group-Graded Algebras

1991 ◽  
Vol 34 (2) ◽  
pp. 224-228
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Group Representation ◽  
Sufficient Conditions ◽  
Projective Cover ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

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.

ON THETA PAIRS FOR A MAXIMAL SUBALGEBRA

2012 ◽  
Vol 11 (01) ◽  
pp. 1250001 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
SARA CHEHRAZI ◽  
SOMAIEH ALIZADEH NIRI
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Nonzero Ideal ◽  
Maximal Subgroups ◽  
Maximal Subalgebra ◽  
Nilpotent Lie Algebras ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Number Of Authors

Given a maximal subalgebra M of a finite-dimensional Lie algebra L, a θ-pair for M is a pair (A, B) of subalgebras such that A ≰ M, B is an ideal of L contained in A ∩ M, and A/B includes properly no nonzero ideal of L/B. This is analogous to the concept of θ-pairs associated to maximal subgroups of a finite group, which has been studied by a number of authors. A θ-pair (A, B) for M is said to be maximal if M has no θ-pair (C, D) such that A < C. In this paper, we obtain some properties of maximal θ-pairs and use them to give some characterizations of solvable, supersolvable and nilpotent Lie algebras.

scholarly journals On the power of a prime dividing the order of the automorphism group of a finite group

1960 ◽  
Vol 4 (4) ◽  
pp. 163-170 ◽  
Author(s):  
J. C. Howarth
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Image Position ◽  
A Finite Group

The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequalityLater Green [2] improved this estimate toIn the Present paper this will be revised, for sufficiently large h, to

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