RATIONAL REPRESENTATIONS AND RATIONAL GROUP ALGEBRA OF VZ $\boldsymbol {p}$-GROUPS

Abstract In this article, we study rational matrix representations of VZ p -groups ( p is any prime). Using our findings on VZ p -groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p -groups of order $\leq p^4$ . Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p -groups and all p -groups of order $\leq p^4$ , ensuring simplicity in the process.