Abstract: In this talk we will discuss the structure of $\mathbb E_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group completion. This is joint work with Maxime Ramzi.