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 ${\mathcal{H}}^2(H,{\mathbb{F}}^{\ast })={\mathsf{r}\mathsf{e}\mathsf{s}}_H^{\mathsf{G}}({\mathcal{H}}^2(\mathsf{G},{\mathbb{F}}^{\ast }))$, where $H$ is a subgroup of $\mathsf{G}$ and ${\mathsf{r}\mathsf{e}\mathsf{s}}_H^{\mathsf{G}}$ is the restriction homomorphism. Posteriorly, we prove that, given any $H_1,H_2\le \mathsf{G}$ and ${\sigma }_i\in {\mathcal{Z}}^2(H_i,{\mathbb{F}}^{\ast })$, $i=1,2$, are equivalent: i) ${\mathbb{F}}^{{\sigma }_1}[H_1]{\mathbb{F}}^{{\sigma }_2}[H_2]$; ii) $H_1\le H_2$ 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, $\mathsf{c}\mathsf{h}\mathsf{a}\mathsf{r}(\mathbb{F})=0$ or $\mathsf{c}\mathsf{h}\mathsf{a}\mathsf{r}(\mathbb{F})$ 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 $\mathfrak{B}\mathfrak{A}$.