Let $\mathbb{ F}$ be a field and $\mathsf{ G}$ a group. This work is inspired in the following problem: "{ \it given a division (simple) $\mathsf{ G}$-graded $\mathbb{ F}$-algebra, is there any other division (simple) $\mathsf{ G}$-graded $\mathbb{ F}$-algebra such that the former can be $\mathsf{ G}$-imbedded in the latter?}". In this work, we answer this question affirmatively for $\mathbb{ F}$ algebraically closed, $\mathsf{ G}$ finite abelian, and associative algebras of finite dimension. To prove this, we apply concepts and properties of Group Cohomology. We show $\mathcal{ H}^2(H, \mathbb{ F}^*)=\mathsf{ res}^\mathsf{ G}_H \left(\mathcal{ H}^2(\mathsf{ G}, \mathbb{ F}^*)\right)$, where $H$ is a subgroup of $\mathsf{ G}$ and $\mathsf{ res}^\mathsf{ G}_H$ is the restriction homomorphism. Posteriorly, we prove that, given any $H_1,H_2\leq\mathsf{ G}$ and $\sigma_i\in\mathcal{ Z}^2(H_i,\mathbb{ F}^*)$, $i=1,2$, are equivalent: i) $\mathbb{ F}^{ \sigma_1}[H_1] \stackrel{ \mathsf{ G}}{ \hookrightarrow} \mathbb{ F}^{ \sigma_2}[H_2]$; ii) $H_1\leq H_2$ and $[\sigma_1]=[\sigma_2]_{ H_1}$; iii) $\mathsf{ T}^{ \mathsf{ G}}(\mathbb{ F}^{ \sigma_2}[H_2])\subseteq \mathsf{ T}^{ \mathsf{ G}}(\mathbb{ F}^{ \sigma_1}[H_1])$, where $\mathsf{ T}^{ \mathsf{ G}}(\mathbb{ F}^{ \sigma_i}[H_i])$ is the $\mathsf{ G}$T-ideal of graded identities of $\mathbb{ F}^{ \sigma_i}[H_i]$. Furthermore, we prove that, given $\mathfrak{ A}$ and $\mathfrak{ B}$ two finite dimensional simple $\mathsf{ G}$-graded $\mathbb{ F}$-algebras, if $\mathbb{ F}$ is algebraically closed, $\mathsf{ char}(\mathbb{ F}) = 0$ or $\mathsf{ char} (\mathbb{ F})$ is coprime with the order of each finite subgroup of $\mathsf{ G}$, and any subgroup of $\mathsf{ G}$ is normal, then $\mathsf{ T}^{ \mathsf{ G}}(\mathfrak{ A})\subseteq \mathsf{ T}^{ \mathsf{ G}}(\mathfrak{ B})$ iff $\mathfrak{ B} \stackrel{ \mathsf{ G}}{ \hookrightarrow} \mathfrak{ A}$.
Categories: math.RA