Homological finiteness conditions for modules over strongly group-graded rings

1996 ◽  
Vol 120 (1) ◽  
pp. 43-54 ◽  
Author(s):  
Jonathan Cornick ◽  
Peter H. Kropholler
Keyword(s):  
Commutative Ring ◽  
Finite Groups ◽  
Group Algebras ◽  
Crossed Products ◽  
Graded Rings ◽  
Finiteness Conditions ◽  
Image Position

Throughout this paper, k denotes a commutative ring. We will develop a theory of homological finiteness conditions for modules over certain graded k-algebras which generalizes known theory for group algebras. The simplest of our results, Theorem A below, generalizes certain results of Aljadeff and Yi on crossed products of polycyclic-by-finite groups (cf. [1, 11]), but also applies to many other crossed products in cases where little was previously known. Before stating the results, we recall definitions of graded and strongly graded rings. Let G be a monoid. Naively, a G-graded k-algebra is a k-algebra R which admits a k-module decomposition,in such a way that Rg Rh ⊆ for all g, h ∈ G. If R is a G-graded k-algebra and X is any subset of G, then we write Rx for the k-submodule of R supported on X; that isNote that if H is a submonoid of G then RH is a subalgebra of R.

scholarly journals The isomorphism problem for group algebras: A criterion

2020 ◽  
Vol 23 (3) ◽  
pp. 435-445
Author(s):  
Taro Sakurai
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Finite Groups ◽  
Isomorphism Problem ◽  
Group Algebras ◽  
Algebra Isomorphism ◽  
Group Isomorphism ◽  
Unital Algebra ◽  
New Class

AbstractLet R be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over R. Our main result states that if G is a hereditary group over R, then a unital algebra isomorphism between group algebras {RG\cong RH} implies a group isomorphism {G\cong H} for every finite group H. As application, we study the modular isomorphism problem, which is the isomorphism problem for finite p-groups over {R=\mathbb{F}_{p}}, where {\mathbb{F}_{p}} is the field of p elements. We prove that a finite p-group G is a hereditary group over {\mathbb{F}_{p}} provided G is abelian, G is of class two and exponent p, or G is of class two and exponent four. These yield new proofs for the theorems by Deskins and Passi–Sehgal.

A Class of Three-Generator, Three-Relation, Finite Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 36-40 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Finite Nilpotent Group ◽  
Image Position

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.The groups presented are defined as:with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.We prove the following result.THEOREM 1. Each of the groups presented is a finite soluble group.We state the following theorem proved by Macdonald (1).THEOREM 2. G1(α, β, 1) is a finite nilpotent group.1. In this section we make some elementary remarks.

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.

FINITENESS CONDITIONS FOR SPECIAL BAND GRADED RINGS

1994 ◽  
Vol 27 (1) ◽  
pp. 171-178
Author(s):  
Andrei V. Kelarev
Keyword(s):  
Graded Rings ◽  
Finiteness Conditions

scholarly journals The exponent of certain finite p-groups

1990 ◽  
Vol 33 (3) ◽  
pp. 483-490 ◽  
Author(s):  
I. O. York
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Group Operation ◽  
Image Position ◽  
Formal Power ◽  
Quotient Groups

In this paper, for R a commutative ring, with identity, of characteristic p, we look at the group G(R) of formal power series with coefficients in R, of the formand the group operation being substitution. The results obtained give the exponent of the quotient groups Gn(R) of this group, n∈ℕ.

scholarly journals Complementation of the Jacobson group in a matrix ring

2005 ◽  
Vol 72 (2) ◽  
pp. 317-324
Author(s):  
David Dolžan
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Matrix Ring ◽  
Graded Rings ◽  
Matrix Rings ◽  
Generators And Relations ◽  
Group Of Units ◽  
The Matrix ◽  
Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

Strong Gelfand pairs of symmetric groups

2020 ◽  
pp. 2150054
Author(s):  
Gradin Anderson ◽  
Stephen P. Humphries ◽  
Nathan Nicholson
Keyword(s):  
Commutative Ring ◽  
Finite Groups ◽  
Symmetric Groups ◽  
Maximal Dimension ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Schur Ring

A strong Gelfand pair is a pair [Formula: see text], of finite groups such that the Schur ring determined by the [Formula: see text]-classes [Formula: see text], is a commutative ring. We find all strong Gelfand pairs [Formula: see text]. We also define an extra strong Gelfand pair [Formula: see text], this being a strong Gelfand pair of maximal dimension, and show that in this case [Formula: see text] must be abelian.

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