One construction of integral representations of $p$-groups and some applications

D. A. Malinin

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 322-338.

Résumé

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.

Détail

Keywords: finite nilpotent groups, integral domain, Dedekind ring, elliptic curves.

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