scholarly journals A remark on the radical of a group algebra

1982 ◽  
Vol 25 (1) ◽  
pp. 69-71 ◽  
Author(s):  
Peter Brockhaus
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Alternative Proof ◽  
Prime Characteristic ◽  
A Finite Group

This paper contains a new proof of a theorem of D. A. R. Wallace [5] in case of a p-solvable group. An alternative proof has been given by K. Motose [4].Let G be a finite group, F a field with prime characteristic p, P a Sylow p-subgroup of G. JFG will denote the Jacobson radical of FG.

scholarly journals Group structure and properties of block ideals of the group algebra

1975 ◽  
Vol 16 (1) ◽  
pp. 22-28 ◽  
Author(s):  
Wolfgang Hamernik
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Group Structure ◽  
Arbitrary Field ◽  
Structure And Properties ◽  
Characteristic P ◽  
Principal Block ◽  
A Finite Group

In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:(i) J(B1) ≤ Z(B1)(ii) J(B1)is commutative,(iii) G is p-nilpotent with abelian Sylowp-subgroups.

scholarly journals On the commutativity of the radical of a group algebra

1965 ◽  
Vol 7 (1) ◽  
pp. 1-8 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Prime Characteristic ◽  
Different Types ◽  
A Finite Group

Over a field of prime characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative.

scholarly journals Remarks on the commutativity of the radicals of group algebras

1979 ◽  
Vol 20 (1) ◽  
pp. 63-68 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Commutator Subgroup ◽  
Group Algebras ◽  
Arbitrary Field ◽  
A Finite Group

Let K be an arbitrary field with characteristic p > 0, G a finite group of order pag′ with (p, g′) = 1, P a p-Sylow subgroup of G and G′ the commutator subgroup of G. For a ring R denote by J(R) the Jacobson radical of R and by Z(R) the centre of R. We write KG for the group algebra of G over K.

On strongly associative group algebras of p-solvable groups

2015 ◽  
Vol 14 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Jonas Gonçalves Lopes
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Group Algebra ◽  
Modular Group ◽  
Solvable Groups ◽  
Crossed Product ◽  
Group Algebras ◽  
Characteristic P ◽  
Partial Action ◽  
A Finite Group

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH *α G. If FH *α G is associative for any G and any α, then FH is called strongly associative. We characterize the strongly associative modular group algebras FH with H being a p-solvable group.

scholarly journals Lower bounds for the radical of the group algebra of a finite p-soluble group

1968 ◽  
Vol 16 (2) ◽  
pp. 127-134 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Lower Bounds ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Soluble Group ◽  
Nilpotent Ideal ◽  
A Finite Group ◽  
Radical Form

Over a field of characteristic p>0 the group algebra of a finite group has a unique maximal nilpotent ideal, the Jacobson radical of the algebra. The powers of the radical form a decreasing and ultimately vanishing series of ideals and it would be of interest to determine the least vanishing power. Apart from the work of Jennings (3) on p-groups little is known in general (cf. (5)) about this particular power of the radical (cf. Remarks of Brauer in (4), p. 144. Problem 15). In this paper we give non-trivial lower bounds for the index of the least vanishing power of the radical when the group is p-soluble. Of the lower bounds we give we show that that lower bound, which is dependent solely on the order of the group, is the best possible such bound and that this bound is invalid if the assumption of p-solubility is omitted.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

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.

scholarly journals A characterization of PGL (2,pn) by some irreducible complex character degrees

2016 ◽  
Vol 99 (113) ◽  
pp. 257-264 ◽  
Author(s):  
Somayeh Heydari ◽  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Group Algebra ◽  
Character Degrees ◽  
Complex Character ◽  
Complex Group ◽  
Complex Group Algebra ◽  
A Finite Group

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2,p) |, p ? cd(G) and max(cd(G)) = p+1, then G ? PGL(2,p), SL(2, p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also, we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G ? PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely determined by the structure of its complex group algebra.

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