Inclusion of regular and linear languages in group languages

Abstract: Let $$\Sigma = X\cup X^{-1} = { x_1 ,x_2 ,..., x_m ,x_1^{-1} ,x_2^{-1} ,..., x_m^{-1} }$$ and let $G$ be a group with set of generators $\Sigma$. Let $\mathfrak{L} (G) =\left{ \left. \omega \in \Sigma^* \; \right\vert \;\omega \equiv e \; (\textrm{mod} \; G) \right} \subseteq \Sigma<sup>*$ be the group language representing $G$, where $\Sigma<sup>*$ is a free monoid over $\Sigma$ and $e$ is the identity in $G$. The problem of determining whether a context-free language is subset of a group language is discussed. Polynomial algorithms are presented for testing whether a regular language, or a linear language is included in a group language. A few finite sets are built, such that each of them is included in the group language $\mathfrak{L} (G)$ if and only if the respective context-free language is included in $\mathfrak{L} (G)$.