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

Abstract

For a partition $\underlineλ = (λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^{ρ_3}>\ldots>λ_{k}^{ρ_k})$ and its associated finite abelian $p$-group $\mathcal{A}_{\underlineλ}=\underset{i=1}{\overset{k}{\oplus}} (\mathbb{Z}/p^{λ_i}\mathbb{Z})^{ρ_i}$, where $p$ is a prime, we consider two actions of its automorphism group $\mathcal{G}_{\underlineλ}$ on $\mathcal{A}_{\underlineλ}$. The first action is the natural action $g\bullet a=\ ^ga$ for all $g\in\mathcal{G}_{\underlineλ}$ and $a\in\mathcal{A}_{\underlineλ}$ where the action map is denoted by $Λ_1=Id_{\mathcal{G}_{\underlineλ}}:\mathcal{G}_{\underlineλ}\longrightarrow \mathcal{G}_{\underlineλ}$ and the second action is the trivial action $g\bullet a=a$ for all $g\in\mathcal{G}_{\underlineλ}$ and $a\in\mathcal{A}_{\underlineλ}$ where the action map is denoted by $Λ_2:\mathcal{G}_{\underlineλ}\longrightarrow \{e\}\subset\mathcal{G}_{\underlineλ}$ the trivial map. For the natural action $Λ_1$, we show that the first and second cohomology groups $H_{Λ_1}^i(\mathcal{G}_{\underlineλ},\mathcal{A}_{\underlineλ}),i=1,2$ vanish for any partition $\underlineλ$ for an odd prime $p$. For the trivial action $Λ_2$ we show that, for an odd prime $p$, the first cohomology group $H_{Λ_2}^1(\mathcal{G}_{\underlineλ},\mathcal{A}_{\underlineλ})$ and for an odd prime $p\neq 3$, the second cohomology group $H_{Λ_2}^2(\mathcal{G}_{\underlineλ},\mathcal{A}_{\underlineλ})$ vanish if and only if the difference between two successive parts of the partition $\underlineλ$ is at most one. This is done by using the $mod\ p$ cohomologies $H^i(\mathcal{G}_{\underlineλ},\mathbb{Z}/p\mathbb{Z}),i=1,2$.