  An original non-standard approach to describing the structure of a column
stabilizer in a group of $n \times n$ matrices over a polynomial ring or a
Laurent polynomial ring of $n$ variables is presented. The stabilizer is
described as an extension of a subgroup of a rather simple structure using the
$(n-1) \times (n-1)$ matrix group of congruence type over the corresponding
ring of $n-1$ variables. In this paper, we consider cases where $n \leq 3.$ For
$n = 2$, the stabilizer is defined as a one-parameter subgroup, and the proof
is carried out by direct calculation. The case $n = 3$ is nontrivial; the
approach mentioned above is applied to it. Corollaries are given to the results
obtained. In particular, we prove that for the stabilizer in the question, it
is not generated by its a finite subset together with the so-called tame
stabilizer of the given column. We are going to study the cases when $n \geq 4$
in a forthcoming paper. Note that a number of key subgroups of the groups of
automorphisms of groups are defined as column stabilizers in matrix groups. For
example, this describes the subgroup IAut($M_r$) of automorphisms that are
identical modulo a commutant of a free metabelian group $M_r$ of rank $r$. This
approach demonstrates the parallelism of theories of groups of automorphisms of
groups and matrix groups that exists for a number of well-known groups. This
allows us to use the results on matrix groups to describe automorphism groups.
In this work, the classical theorems of Suslin, Cohn, as well as Bachmuth and
Mochizuki are used.
