2002, VOLUME 8, NUMBER 2, PAGES 407-473

Fully invariant subgroups of Abelian groups and full transitivity

S. Ya. Grinshpon


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An Abelian group A is said to be fully transitive if for any elements a,b Î A with H(a) £ H(b) (H(a)H(b) are the height-matrices of elements a and b) there exists an endomorphism of A sending a into b. We say that an Abelian group A is H-group if any fully invariant subgroup S of A has the form S = {a Î A | H(a) ³ M}, where M is some w ´ w-matrix with ordinal numbers and symbol ¥ for entries. The description of fully transitive groups and H-groups in various classes of Abelian groups is obtained. The results of this paper show that every H-group is a fully transitive group, but there are fully transitive torsion free groups and mixed groups, which are not H-groups. The full description of fully invariant subgroups and their lattice for fully transitive groups in various classes of Abelian groups is obtained.

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Last modified: November 26, 2002