FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

S. Ya. Grinshpon

Abstract

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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