Displaying similar documents to “Basic subgroups in modular abelian group algebras”

Ulm-Kaplansky invariants of S(KG)/G

P. V. Danchev (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let G be an infinite abelian p-group and let K be a field of the first kind with respect to p of characteristic different from p such that s p ( K ) = or s p ( K ) = 0 . The main result of the paper is the computation of the Ulm-Kaplansky functions of the factor group S(KG)/G of the normalized Sylow p-subgroup S(KG) in the group ring KG modulo G. We also characterize the basic subgroups of S(KG)/G by proving that they are isomorphic to S(KB)/B, where B is a basic subgroup of G.

The generalized criterion of Dieudonné for valuated p -groups

Peter Vassilev Danchev (2006)

Acta Mathematica Universitatis Ostraviensis

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We prove that if G is an abelian p -group with a nice subgroup A so that G / A is a Σ -group, then G is a Σ -group if and only if A is a Σ -subgroup in G provided that A is equipped with a valuation induced by the restricted height function on G . In particular, if in addition A is pure in G , G is a Σ -group precisely when A is a Σ -group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc....

Basic subgroups in commutative modular group rings

Peter Vassilev Danchev (2004)

Mathematica Bohemica

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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...

Commutative modular group algebras of p -mixed and p -splitting abelian Σ -groups

Peter Vassilev Danchev (2002)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a p -mixed abelian group and R is a commutative perfect integral domain of char R = p > 0 . Then, the first main result is that the group of all normalized invertible elements V ( R G ) is a Σ -group if and only if G is a Σ -group. In particular, the second central result is that if G is a Σ -group, the R -algebras isomorphism R A R G between the group algebras R A and R G for an arbitrary but fixed group A implies A is a p -mixed abelian Σ -group and even more that the high subgroups of A and G are isomorphic, namely,...

Isolated subgroups of finite abelian groups

Marius Tărnăuceanu (2022)

Czechoslovak Mathematical Journal

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We say that a subgroup H is isolated in a group G if for every x G we have either x H or x H = 1 . We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.