  Let $\mathfrak A_n$ be the subset of the standard integer lattice $\mathbb
Z^n$, $\mathfrak A_n\subset\mathbb Z^n$ which is defined by the condition
$\mathfrak A_n=((a_1,...,a_n)\in\mathbb Z^n | a_i\not\equiv a_j\mod n, \forall
i,j\in {1,... n})$. It is clear that the standard addition on the lattice
$\mathbb Z^n$ does not induce the group structure on the set $\mathfrak A_n$
since the componentwise sum of some two vectors may contain components which
are equal modulo $n$. Our aim is to find a new associative multiplication on
the lattice $\mathbb Z^n$ such that the induced multiplication on the set
$\mathfrak A_n$ gives it the group structure. In this paper the group structure
on the subset $\mathfrak A_n$ of the integer lattice $\mathbb Z^n$ is studied
by means of the constructions of a deformation of a group multiplication. The
geometric realization of this group in the enveloping space and its generators
and relations between them are found. We begin with the main constructions and
the results we need for them.
