Yurchenko N. On finite subgroups of the general linear group over integral domains

Let R be a principal ideal domain of the characteristic zero and р is not invertible element of R. The problem of the conjugation of Sylow р-subgroups of the general linear group over R has been solved in the manuscript. Depended on R and р sufficient conditions of the isomorphism of Sylow р-subgroups of the general linear group over the principal ideal domain R of the characteristic zero have been founded. The conditions of existence of some Sylow р-subgroups of the group GL(р, R), where R is the ring of integers of the finite extension of the field of р-adic numbers such that containes primitive рn-th root of identity, have been founded. Let R be the ring of all algebraic integers. It has been proved that there are infinity many pairwise nonisomorphic Sylow р-subgroups of the general linear group GL(n, R) (n >1). All minimal irreducible р-subgroups of the groups GL(pr, Q(e)) (r ? 2) (ep = 1, e ? 1, p is prime) and GL(рs(p - 1), Q) (s ? 2) have been described. Some classes of minimal irreducible solvable subgroups of the group GL(pq, Q), where p and q are primes (p > q) and q divides р - 1, have been founded.