Some well-known classical results related to the description of integral representations
of finite groups over Dedekind rings $R$, especially for the rings
of integers $\mathbf{Z}$ and $p$-adic integers $\mathbf{Z}_p$ and maximal orders of local fields and fields of
algebraic numbers go back to classical papers by S. S. Ryshkov, P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the
representations, and one of the possible approaches is to describe maximal finite subgroups of $GL_n(R)$
over Dedekind rings $R$ for a fixed positive integer $n$.
The basic idea underlying a geometric approach was
given in Ryshkov’s papers on the computation of the finite subgroups
of $GL_n(\mathbf{Z})$ and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under
the extension of the Dedekind rings $R$ in general, and in what way the representations of arbitrary $p
$-groups, supersolvable groups or groups of a given nilpotency class can be approached.
In the
present paper the above classes of groups are treated, in particular, it is proven that for a fixed $n$ and
any given nonabelian $p$-group $G$ there is an infinite
number of pairwise non-isomorphic absolutely irreducible representations of the group $G$. A
combinatorial construction of the series of these representations is given explicitly.
In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible
representations of finite $p$-groups with the extra congruence conditions is constructed.
We
consider certain related questions including the embedding problem in Galois theory for local
faithful primitive representations of supersolvable groups and integral representations arising from
elliptic curves.
Bibliography: 27 titles.
