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 $a,b \in A$ with $\mathbb H(a) \leq \mathbb H(b)$ ($\mathbb H(a)$, $\mathbb H(b)$ 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 $\mathbb H$-group if any fully invariant subgroup $S$ of $A$ has the form $S = \{a \in A \mid \mathbb H(a) \geq M\}$, where $M$ is some $\omega \times \omega$-matrix with ordinal numbers and symbol $\infty$ for entries. The description of fully transitive groups and $\mathbb H$-groups in various classes of Abelian groups is obtained. The results of this paper show that every $\mathbb H$-group is a fully transitive group, but there are fully transitive torsion free groups and mixed groups, which are not $\mathbb H$-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