Integral representations of Zp and fixed point sets of p-group actions on tori

Let p and q be different prime numbers and let aZ⊕a1⊕⋯⊕ab⊕E1⊕⋯⊕Ec be the Diedrichsen–Reiner decomposition of an integral representation of Zp. We show that a, b, and c can be found solving a system of appropriate linear equations in fields Zp and Zq. We prove that solving similar equations one can calculate the number of connected components of p-group actions on tori.