Subgroup rigidity in finite-dimensional group algebras over $p$-groups

Abstract:

In 1986, Roggenkamp and Scott proved in [RS1] Theorem 1.1. Let $G$ be a finite $p$-group for some prime $p$, and $S$ a local or semilocal Dedekind domain of characteristic $0$ with a unique maximal ideal containing $p$ (for example, $S = {\mathbb {Z}_p}$ where ${\mathbb {Z}_p}$ is the $p$-adic integers). If $H$ is a subgroup of the normalized units of $SG$ with $|H| = |G|$, then $H$ is conjugate to $G$ by an inner automorphism of $SG$. In the Appendix of a later paper [S], Scott outlined a possible proof of a related result: Theorem 1.3. Let $S$ be a complete, discrete valuation domain of characteristic $0$ having maximal ideal $\wp$ and residue field $F \cong S/\wp$ of characteristic $p$. Let $G$ be a finite $p$-group, and let $U$ be a finite group of normalized units in $SG$. Then there is a unit $w$ in $SG$ such that $wU{w^{ - 1}} \leq G$. The author later filled in that outline to give a complete proof of Theorem 1.3 and, at the urging of Scott, has been able to extend that result to Theorem 1.2. Let $S$ be a complete, discrete valuation ring of characteristic $0$ having maximal ideal $\wp$ containing $p$. Let $A$ be a local $S$-algebra that is finitely generated as an $S$-module, and let $G$ be a finite $p$-group. Then any finite, normalized subgroup of the $S$-algebra $\mathcal {A} = A{ \otimes _S}SG$ is conjugate to a subgroup of $G$.