Graded Imbeddings in Finite Dimensional Simple Graded Algebras

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 $\mathfrak{B} \stackrel{\mathsf{G}}{\hookrightarrow} \mathfrak{A}$.