Dominions of torsion free abelian groups in metabelian groups
A. Budkin
The dominion of a subalgebra \(H\) in a universal
algebra \(A\) (in a class \(\mathcal{M}\)) is the set of all
elements \(a\in A\) such that, for all homomorphisms
\(f,g:A\rightarrow B\in \mathcal{M}\), if \(f,g\) coincide on
\(H\) then \(a^f=a^g\).
We write
\[{\rm dom}_A^{\mathcal{M}}(H)=\{ a\in A\mid \forall M\in
{\mathcal{M}},\forall f,g:A\rightarrow M,\ \
\mbox{if}\ f\mid _H=g\mid _H \ \mbox{then} \ a^f=a^g\}
\]
Here, as usual, \(f\mid _H\) denotes the restriction
of \(f\) on \(H\).
The concept of dominions was introduced by J. R.
Isbell (1965) to study epimorphisms.
Later, dominions were
investigated in several classes of algebras (G. M.
Bergman 1990; D. Wasserman 2001; H. E. Scheiblich
1976; B. Mitchell 1972; D. Saracino 1983; A. Magidin
1999 (9 papers); S. Shakhova 2005, 2006, 2010).
There is a connection between dominions and amalgams.
See the survey article by P. M. Higgins (1988) for
the details.
It is not hard
to see that \({\rm dom}_A^{\mathcal{M}}( - )\) is a closure
operator on the lattice of subalgebras of \(A\) in the
sense that it is extensive (the dominion of \(H\)
contains \(H\)), idempotent (the dominion of the
dominion of \(H\) equals the dominion of \(H\)), and
isotone (if \(H\subset K\), then the dominion of \(H\) is
contained in that of \(K\)).
The notion of a closed subgroup arose.
We say that a group \(H\) is closed (or absolutely closed) in a class \( \mathcal{M}\) of groups
if, for every \(G\in \mathcal{M}\) and every embedding \(H\leq G\), we have
\[ {\rm dom}^{ \mathcal{M}}_{G}(H)=H.\]
In this paper, we study closed groups in the variety \(\mathcal{A}^2\) of metabelian groups.
Theorem 1.
Every nontrivial torsion free abelian group is not closed in the class of metabelian groups.
Theorem 2.
Let a group \(H\) be a torsion free group, let \(G\) be a metabelian group, and let \(H\leq G\).
If
\(H\cap G'=(1)\)
then
\( {\rm dom}^{ \mathcal{A}^2}_{G}(H)=H\).
