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

Abstract

For a partition $\underline{\lambda }=({\lambda }_1^{{\rho }_1}>{\lambda }_2^{{\rho }_2}>{\lambda }_3^{{\rho }_3}>\dots >{\lambda }_k^{{\rho }_k})$ and its associated finite abelian $p$-group ${\mathcal{A}}_{\underline{\lambda }}=\underset{i=1}{\stackrel{k}{\oplus }}(\mathbb{Z}/p^{{\lambda }_i}\mathbb{Z}{)}^{{\rho }_i}$, 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 $g\bullet a={}^{g}a$ for all $g\in {\mathcal{G}}_{\underline{\lambda }}$ and $a\in {\mathcal{A}}_{\underline{\lambda }}$ where the action map is denoted by ${\Lambda }_1=I{d}_{{\mathcal{G}}_{\underline{\lambda }}}:{\mathcal{G}}_{\underline{\lambda }}⟶{\mathcal{G}}_{\underline{\lambda }}$ 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 ${\Lambda }_2:{\mathcal{G}}_{\underline{\lambda }}⟶\{e\}\subset {\mathcal{G}}_{\underline{\lambda }}$ 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 $p\ne 3$, 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 $modp$ cohomologies $H^i({\mathcal{G}}_{\underline{\lambda }},\mathbb{Z}/p\mathbb{Z}),i=1,2$.

arXiv comment62 pages, 1 figure