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Basic subgroups in commutative modular group rings

Peter Vassilev Danchev (2004)

Mathematica Bohemica

Let S ( R G ) be a normed Sylow p -subgroup in a group ring R G of an abelian group G with p -component G p and a p -basic subgroup B over a commutative unitary ring R with prime characteristic p . The first central result is that 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) is basic in S ( R G ) and B [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] is p -basic in V ( R G ) , and [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] G p / G p is basic in S ( R G ) / G p and [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] G / G is p -basic in V ( R G ) / G , provided in both cases G / G p is p -divisible and R is such that its maximal perfect subring R p i has no nilpotents whenever i is natural. The second major result is that B ( 1 + I ( R G ; B p ) ) is p -basic in V ( R G ) and ( 1 + I ( R G ; B p ) ) G / G is p -basic in V ( R G ) / G ,...

Basic subgroups in modular abelian group algebras

Peter Vassilev Danchev (2007)

Czechoslovak Mathematical Journal

Suppose F is a perfect field of c h a r F = p 0 and G is an arbitrary abelian multiplicative group with a p -basic subgroup B and p -component G p . Let F G be the group algebra with normed group of all units V ( F G ) and its Sylow p -subgroup S ( F G ) , and let I p ( F G ; B ) be the nilradical of the relative augmentation ideal I ( F G ; B ) of F G with respect to B . The main results that motivate this article are that 1 + I p ( F G ; B ) is basic in S ( F G ) , and B ( 1 + I p ( F G ; B ) ) is p -basic in V ( F G ) provided G is p -mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...

Butler groups and Shelah's Singular Compactness

Ladislav Bican (1996)

Commentationes Mathematicae Universitatis Carolinae

A torsion-free group is a B 2 -group if and only if it has an axiom-3 family of decent subgroups such that each member of has such a family, too. Such a family is called S L 0 -family. Further, a version of Shelah’s Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group B in a prebalanced and TEP exact sequence 0 K C B 0 is a B 2 -group provided K and C are so.

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