On the Vanishing Criterion for the Cohomology Groups of the Automorphism Group of a finite Abelian $p$-Group

TL;DR

This paper investigates the vanishing of certain cohomology groups of automorphism groups of finite abelian p-groups under different actions, revealing conditions under which these groups vanish, especially for odd primes.

Contribution

It provides new criteria for the vanishing of cohomology groups of automorphism groups of finite abelian p-groups under natural and trivial actions, using mod p cohomology techniques.

Findings

Cohomology groups vanish for natural action at odd primes.

Vanishing of first and second cohomology under trivial action depends on partition differences.

Results apply specifically for odd primes, with special case for p ≠ 3.

Abstract

For a partition $\underline{\lambda} = (\lambda_{1}^{\rho_1}>\lambda_{2}^{\rho_2}>\lambda_{3}^{\rho_3}>\ldots>\lambda_{k}^{\rho_k})$ and its associated finite abelian $p$-group $\mathcal{A}_{\underline{\lambda}}=\underset{i=1}{\overset{k}{\oplus}} (\mathbb{Z}/p^{\lambda_i}\mathbb{Z})^{\rho_i}$, where $p$ is a prime, we consider two actions of its automorphism group $\mathcal{G}_{\underline{\lambda}}$ on $\mathcal{A}_{\underline{\lambda}}$. The first action is the natural action $g\bullet a=\ ^ga$ for all $g\in\mathcal{G}_{\underline{\lambda}}$ and $a\in\mathcal{A}_{\underline{\lambda}}$ where the action map is denoted by $\Lambda_1=Id_{\mathcal{G}_{\underline{\lambda}}}:\mathcal{G}_{\underline{\lambda}}\longrightarrow \mathcal{G}_{\underline{\lambda}}$ and the second action is the trivial action $g\bullet a=a$ for all $g\in\mathcal{G}_{\underline{\lambda}}$ and $a\in\mathcal{A}_{\underline{\lambda}}$ where the action map is denoted by $\Lambda_2:\mathcal{G}_{\underline{\lambda}}\longrightarrow \{e\}\subset\mathcal{G}_{\underline{\lambda}}$ the trivial map. For the natural action $\Lambda_1$, we show that the first and second cohomology groups $H_{\Lambda_1}^i(\mathcal{G}_{\underline{\lambda}},\mathcal{A}_{\underline{\lambda}}),i=1,2$ vanish for any partition $\underline{\lambda}$ for an odd prime $p$. For the trivial action $\Lambda_2$ we show that, for an odd prime $p$, the first cohomology group $H_{\Lambda_2}^1(\mathcal{G}_{\underline{\lambda}},\mathcal{A}_{\underline{\lambda}})$ and for an odd prime $p\neq 3$, the second cohomology group $H_{\Lambda_2}^2(\mathcal{G}_{\underline{\lambda}},\mathcal{A}_{\underline{\lambda}})$ vanish if and only if the difference between two successive parts of the partition $\underline{\lambda}$ is at most one. This is done by using the $mod\ p$ cohomologies $H^i(\mathcal{G}_{\underline{\lambda}},\mathbb{Z}/p\mathbb{Z}),i=1,2$.