Classifying the Groups of Order $p^3$ in Lean

This note discusses our formalisation in Lean 4 of the classification of groups of order $p^3$ for a prime number $p$, using mathlib4. We present the five isomorphism classes and give a detailed account of the formalisation, with particular emphasis on the non-abelian case, which requiring the most substantial formal development. For odd~$p$, the non-abelian groups are the Heisenberg group $\text{Heis}(\mathbb{Z}\mathrm{/}p\mathbb{Z})$ and the semidirect product $\mathbb{Z}\mathrm{/}{p}^2\mathbb{Z}⋊\mathbb{Z}\mathrm{/}p\mathbb{Z}$; for $p=2$, they are $D_4$ and $Q_8$. We describe the construction of these concrete groups, the structural lemmas about centers, commutators, and exponents, and the explicit isomorphism constructions that classify an arbitrary non-abelian $p^3$-group.