Let $G$ be a finite abelian group viewed a $\mathbb{Z}$-module and let $\mathcal{G} = (V, E)$ be a simple graph. In this paper, we consider a graph $Γ(G)$ called as a \textit{group-annihilator} graph. The vertices of $Γ(G)$ are all elements of $G$ and two distinct vertices $x$ and $y$ are adjacent in $Γ(G)$ if and only if $[x : G][y : G]G = \{0\}$, where $x, y\in G$ and $[x : G] = \{r\in\mathbb{Z} : rG \subseteq \mathbb{Z}x\}$ is an ideal of a ring $\mathbb{Z}$. We discuss in detail the graph structure realised by the group $G$. Moreover, we study the creation sequence, hyperenergeticity and hypoenergeticity of group-annihilator graphs. Finally, we conclude the paper with a discussion on Laplacian eigen values of the group-annhilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action: $Aut(Γ(G)) \times G \rightarrow G$.