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Summary:
Let $G$ be a finite abelian group with identity element $1_G$ and $L=\bigoplus _{g\in G}L^g$ be an infinite dimensional $G$-homogeneous vector space over a field of characteristic $0$. Let $E=E(L)$ be the Grassmann algebra generated by $L$. It follows that $E$ is a $G$-graded algebra. Let $|G|$ be odd, then we prove that in order to describe any ideal of $G$-graded identities of $E$ it is sufficient to deal with $G^{\prime }$-grading, where $|G^{\prime }| \le |G|$, $\dim _FL^{1_{G^{\prime }}}=\infty$ and $\dim _FL^{g^{\prime }}<\infty$ if $g^{\prime }\ne 1_{G^{\prime }}$. In the same spirit of the case $|G|$ odd, if $|G|$ is even it is sufficient to study only those $G$-gradings such that $\dim _FL^g=\infty$, where $o(g)=2$, and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of $E$ in the case $\dim L^{1_G}=\infty$ and $\dim L^g<\infty$ if $g\ne 1_G$.
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