scholarly journals Skew group rings and maximal orders

1995 ◽  
Vol 37 (2) ◽  
pp. 249-263 ◽  
Author(s):  
R. Martin
Keyword(s):  
Finite Group ◽  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Maximal Order ◽  
Group Rings ◽  
Maximal Orders ◽  
Noetherian Domain ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

Let S be a prime Noetherian ring and G a finite group acting on 5 such that Gis x-outer on S. We give sufficient conditions for the skew group ring S * Gto be a prime maximal order. If we impose the further hypothesis that the order of Gbe a unit of S, then these conditions are also necessary. Moreover, if S is a commutative Noetherian domain, then there are necessary and sufficient conditions for S*Gto be a prime maximal order, without requiring that the order of G be a unit in S.

Free Subgroups of Units in Group Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 309-312 ◽  
Author(s):  
Jairo Zacarias Gonçalves
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Free Subgroup ◽  
Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Free Subgroups

AbstractIn this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.Some extensions of this results to infinite nilpotent and FC groups are also considered.

scholarly journals Skew group rings which are Galois

2000 ◽  
Vol 23 (4) ◽  
pp. 279-283
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Extension ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inner Automorphism ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

LetS*Gbe a skew group ring of a finite groupGover a ringS. It is shown that ifS*Gis anG′-Galois extension of(S*G)G′, whereG′is the inner automorphism group ofS*Ginduced by the elements inG, thenSis aG-Galois extension ofSG. A necessary and sufficient condition is also given for the commutator subring of(S*G)G′inS*Gto be a Galois extension, where(S*G)G′is the subring of the elements fixed under each element inG′.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

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.

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.

GROUP ALGEBRAS WITH UNIT GROUPS OF DERIVED LENGTH THREE

2010 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
HARISH CHANDRA ◽  
MEENA SAHAI
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
Characteristic P ◽  
Derived Length ◽  
A Finite Group ◽  
Necessary And Sufficient

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ3(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained.

The Influence of Generalized Frattini Subgroups on the Solvability of a Finite Group

1970 ◽  
Vol 22 (1) ◽  
pp. 41-46 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Proper Subgroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fitting Subgroup ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.

scholarly journals Group Algebras With Central Radicals

1962 ◽  
Vol 5 (3) ◽  
pp. 103-108 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Algebras ◽  
A Finite Group ◽  
Necessary And Sufficient

It is well known that when the characteristic p(≠ 0) of a field divides the order of a finite group, the group algebra possesses a non-trivial radical and that, if p does not divide the order of the group, the group algebra is semi-simple. A group algebra has a centre, a basis for which consists of the class-sums. The radical may be contained in this centre; we obtain necessary and sufficient conditions for this to happen.

scholarly journals Simple semigroup graded rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550102 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Simple Semigroup ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Crossed Products ◽  
Graded Rings ◽  
Unital Ring ◽  
Nonzero Idempotent ◽  
Hypercentral Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and Re has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring f ReGe f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Gonçalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

scholarly journals Baer and quasi-Baer properties of group rings

2007 ◽  
Vol 83 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Zhong Yi ◽  
Yiqiang Zhou
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Baer Group ◽  
Baer Ring ◽  
Fixed Ring ◽  
A Finite Group ◽  
Left Annihilator

AbstractA ring R is said to be a Baer (respectively, quasi-Baer) ring if the left annihilator of any nonempty subset (respectively, any ideal) of R is generated by an idempotent. It is first proved that for a ring R and a group G, if a group ring RG is (quasi-) Baer then so is R; if in addition G is finite then |G|–1 € R. Counter examples are then given to answer Hirano's question which asks whether the group ring RG is (quasi-) Baer if R is (quasi-) Baer and G is a finite group with |G|–1 € R. Further, efforts have been made towards answering the question of when the group ring RG of a finite group G is (quasi-) Baer, and various (quasi-) Baer group rings are identified. For the case where G is a group acting on R as automorphisms, some sufficient conditions are given for the fixed ring RG to be Baer.

On c#-normal subgroups of finite groups

2016 ◽  
Vol 16 (08) ◽  
pp. 1750160
Author(s):  
Guo Zhong ◽  
Shi-Xun Lin
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Necessary And Sufficient

Let [Formula: see text] be a subgroup of a finite group [Formula: see text]. We say that [Formula: see text] is a [Formula: see text]-normal subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a [Formula: see text]-subgroup of [Formula: see text]. In the present paper, we use [Formula: see text]-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be [Formula: see text]-supersolvable, [Formula: see text]-nilpotent and solvable. Our results extend and improve some recent ones.

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