Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra
Year: 2026
Type: preprint
Abstract: Article submitted to Bull. London Math. Soc.Abstract.Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\operatorname{Out}(H)$ and isomorphism classes of the associated small group rings with orbits of $\operatorname{Out}(\mathbb{F}_p H)$. The discrepancy between the two is measured by the group $U(H) \le GL(V)$ of displacements induced by the normalized unipotent automorphisms of $\mathbb{F}_p H$; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for $H$ of exponent $p$ and class at most $p-2$ every such automorphism acts on $V$ exactly as an automorphism of $H$ does, and that $U(H) = \exp \Omega \bigl( \operatorname{Der}_{>0}(\mathfrak{l}(H)) \bigr)$, where $\mathfrak{l}(H)$ is the Lazard Lie algebra of $H$, $\operatorname{Der}_{>0}$ its derivations raising the lower central filtration, and $\Omega$ the natural action of derivations on $H_2(\mathfrak{l}(H))$. Hence $U(H)=1$ precisely when every derivation of positive degree acts trivially on $H_2(\mathfrak{l}(H))$. We compute $U(H)$ for all groups of class $2$ and exponent $p$, obtaining $p^{2n(n-1)}$ for the Heisenberg groups $\operatorname{Heis}(p^n)$.
Source: arXiv (Cornell University)
Author: Tuzzolo, Diego
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Universidade de São Paulo
OpenAlex ID: W7213263786