Rational Representations and Rational Group Algebra of VZ p-groups

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