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Let be a normed Sylow -subgroup in a group ring of an abelian group with -component and a -basic subgroup over a commutative unitary ring with prime characteristic . The first central result is that is basic in and is -basic in , and is basic in and is -basic in , provided in both cases is -divisible and is such that its maximal perfect subring has no nilpotents whenever is natural. The second major result is that is -basic in and is -basic in ,...
Suppose is a perfect field of and is an arbitrary abelian multiplicative group with a -basic subgroup and -component . Let be the group algebra with normed group of all units and its Sylow -subgroup , and let be the nilradical of the relative augmentation ideal of with respect to . The main results that motivate this article are that is basic in , and is -basic in provided is -mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...
A torsion-free group is a -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 -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 in a prebalanced and TEP exact sequence is a -group provided and are so.
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