On the representation dimension of finite $p$-groups

Abstract

For a finite group $G$, the representation dimension $δ(G)$ is the least dimension of a faithful complex representation of $G$. We prove that $δ(H\times K) = δ(H) + δ(K)$ whenever $H$ and $K$ are $p$-groups for a fixed prime $p$, and show by example that this additivity can fail more generally for nilpotent groups. We also determine $δ(G)$ for several important classes of non-abelian $p$-groups, {\it viz.} VZ groups, Camina groups, and metacyclic groups; and compute $δ(G)$ for all groups of order $p^6$ ($p\geq 5$), organized by their isoclinism family.