Integral representations with trivial first cohomology groups

Author(s):

Shizuo Endo ◽

Takehiko Miyata

Keyword(s):

Finite Group ◽

Equivalence Relation ◽

Integral Representations ◽

Cohomology Groups ◽

Finitely Generated ◽

Abelian Semigroup ◽

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).

On a classification of the function fields of algebraic tori

Nagoya Mathematical Journal ◽

10.1017/s0027763000016408 ◽

1975 ◽

Vol 56 ◽

pp. 85-104 ◽

Cited By ~ 34

Author(s):

Shizuo Endo ◽

Takehiko Miyata

Keyword(s):

Finite Group ◽

Equivalence Relation ◽

Galois Extension ◽

Function Fields ◽

Equivalence Classes ◽

Finitely Generated ◽

Abelian Semigroup ◽

Algebraic Tori ◽

Let II be a finite group and denote by MII the class of all (finitely generated Z-free) II-modules. In the previous paper [3] we defined an equivalence relation in MII and constructed the abelian semigroup T(II) by giving an addition to the set of all equivalence classes in MII. The investigation of the semigroup T(II) seems interesting and important, because this gives a classification of the function fields of algebraic tori defined over a field k which split over a Galois extension of k with group II.

A note on the fixed subring of an FPF ring

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003543 ◽

1989 ◽

Vol 40 (1) ◽

pp. 109-111 ◽

Cited By ~ 3

Author(s):

John Clark

Keyword(s):

Finite Group ◽

Associative Ring ◽

Finitely Generated ◽

Group Of Automorphisms ◽

Direct Sum ◽

Epimorphic Image ◽

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Inertial subalgebras of algebras possessing finite automorphism groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700012295 ◽

1979 ◽

Vol 28 (3) ◽

pp. 335-345 ◽

Author(s):

Nicholas S. Ford

Keyword(s):

Finite Group ◽

Commutative Ring ◽

Jacobson Radical ◽

Automorphism Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Finitely Generated ◽

Fixed Ring ◽

A Finite Group ◽

Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

The mod ℭ Suspension Theorem

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-078-7 ◽

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 ◽

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.

Simple groups and irreducible lattices in wreath products

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.5 ◽

2020 ◽

pp. 1-12 ◽

Author(s):

ADRIEN LE BOUDEC

Keyword(s):

Finite Group ◽

Wreath Product ◽

Free Group ◽

Wreath Products ◽

Simple Groups ◽

Finitely Generated ◽

Locally Compact ◽

Regular Tree ◽

Finitely Generated Groups ◽

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

On locally nilpotent groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100030905 ◽

1956 ◽

Vol 52 (1) ◽

pp. 5-11 ◽

Cited By ~ 23

Author(s):

D. H. McLain ◽

P. Hall

Keyword(s):

Finite Group ◽

Nilpotent Group ◽

Direct Product ◽

Finite Subset ◽

Nilpotent Groups ◽

Finitely Generated ◽

Locally Finite ◽

Locally Nilpotent Groups ◽

Locally Nilpotent ◽

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

Explicit homotopy equivalences in dimension two

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004102006011 ◽

2002 ◽

Vol 133 (3) ◽

pp. 411-430 ◽

Cited By ~ 6

Author(s):

F. E. A. JOHNSON

Keyword(s):

Finite Group ◽

Exact Sequence ◽

Finitely Generated ◽

Stable Class ◽

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).

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

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500033050 ◽

1954 ◽

Vol 2 (2) ◽

pp. 66-76 ◽

Cited By ~ 10

Author(s):

Iain T. Adamson

Keyword(s):

Finite Group ◽

Exact Sequence ◽

Cohomology Group ◽

Cohomology Theory ◽

Normal Subgroups ◽

Cohomology Groups ◽

Coset Space ◽

Image Position ◽

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.

ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972709000951 ◽

2010 ◽

Vol 81 (2) ◽

pp. 317-328 ◽

Cited By ~ 4

Author(s):

MARCEL HERZOG ◽

PATRIZIA LONGOBARDI ◽

MERCEDE MAJ

Keyword(s):

Finite Group ◽

Finite Groups ◽

Connected Graph ◽

Solvable Groups ◽

Maximal Subgroups ◽

Finite Solvable Group ◽

Finitely Generated ◽

Infinite Case ◽

Locally Graded Group ◽

AbstractLet G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

Homotopy Formulas for Cyclic Groups Acting on Rings

Canadian Mathematical Bulletin ◽

10.4153/cmb-2008-010-3 ◽

2008 ◽

Vol 51 (1) ◽

pp. 81-85

Author(s):

Christian Kassel

Keyword(s):

Finite Group ◽

Cyclic Group ◽

Cohomology Groups ◽

Joint Work ◽

Cyclic Groups ◽

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.