ON THE COHOMOLOGY GROUPS OF A FINITE GROUP

1955 ◽  
Vol 6 (1) ◽  
pp. 34-47
Author(s):  
ANNE P. COBBE
Keyword(s):  
Finite Group ◽  
Cohomology Groups ◽  
A Finite Group

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 Integral representations with trivial first cohomology groups

1982 ◽  
Vol 85 ◽  
pp. 231-240 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Finite Group ◽  
Equivalence Relation ◽  
Integral Representations ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Abelian Semigroup ◽  
A Finite Group

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

scholarly journals Homotopy Formulas for Cyclic Groups Acting on Rings

2008 ◽  
Vol 51 (1) ◽  
pp. 81-85
Author(s):  
Christian Kassel
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Cohomology Groups ◽  
Joint Work ◽  
Cyclic Groups ◽  
A Finite Group

AbstractThe positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

scholarly journals Cohomology of complements of toric arrangements associated with root systems

2022 ◽  
Vol 9 (1) ◽  
Author(s):  
Olof Bergvall
Keyword(s):  
Finite Group ◽  
Root Systems ◽  
Important Application ◽  
Weyl Groups ◽  
Cohomology Groups ◽  
A Finite Group

AbstractWe develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group $$\Gamma $$ Γ is acting on the arrangement, the algorithm determines the cohomology groups as representations of $$\Gamma $$ Γ . As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

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.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

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