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A direct factor theorem for commutative group algebras

William Ullery (1992)

Commentationes Mathematicae Universitatis Carolinae

Suppose F is a field of characteristic p 0 and H is a p -primary abelian A -group. It is shown that H is a direct factor of the group of units of the group algebra F H .

A note on group algebras of p -primary abelian groups

William Ullery (1995)

Commentationes Mathematicae Universitatis Carolinae

Suppose p is a prime number and R is a commutative ring with unity of characteristic 0 in which p is not a unit. Assume that G and H are p -primary abelian groups such that the respective group algebras R G and R H are R -isomorphic. Under certain restrictions on the ideal structure of R , it is shown that G and H are isomorphic.

A note on normal generation and generation of groups

Andreas Thom (2015)

Communications in Mathematics

In this note we study sets of normal generators of finitely presented residually p -finite groups. We show that if an infinite, finitely presented, residually p -finite group G is normally generated by g 1 , , g k with order n 1 , , n k { 1 , 2 , } { } , then β 1 ( 2 ) ( G ) k - 1 - i = 1 k 1 n i , where β 1 ( 2 ) ( G ) denotes the first 2 -Betti number of G . We also show that any k -generated group with β 1 ( 2 ) ( G ) k - 1 - ε must have girth greater than or equal 1 / ε .

A note on semilocal group rings

Angelina Y. M. Chin (2002)

Czechoslovak Mathematical Journal

Let R be an associative ring with identity and let J ( R ) denote the Jacobson radical of R . R is said to be semilocal if R / J ( R ) is Artinian. In this paper we give necessary and sufficient conditions for the group ring R G , where G is an abelian group, to be semilocal.

An analogue of the Duistermaat-van der Kallen theorem for group algebras

Wenhua Zhao, Roel Willems (2012)

Open Mathematics

Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen...

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