of ${\mathrm{Mon}}$. Then $G$ acts naturally on $K[X]$ by means of homomorphisms. A [*$G$-orbit*]{} is a set of the form $Gz=\{\pi z \mid \pi  \in G\}$, where $z$ is in a set on which $G$ acts. The fact that $G$ acts by monoid homomorphisms on ${\mathrm{Mon}}$ implies that the ideal generated by a union of $G$-orbits in $K[X]$ is automatically [*$G$-stable*]{}, that is, closed under the action of $G$.

We use the notation ${\mathrm{lm}}(f)$ for the [*leading monomial*]{} of $f$, i.e., the $\leq$-largest monomial having non-zero coefficient in $f$. By the requirement that $G$ preserve the order, we have ${\mathrm{lm}}(\pi
f)=\pi\,{\mathrm{lm}}(f)$. Given an ideal $I$ of $K[X]$, ${\mathrm{lm}}(I)$ is an ideal in the monoid ${\mathrm{Mon}}$, that is, ${\mathrm{lm}}(I)$ is closed under multiplication with any element from ${\mathrm{Mon}}$. If $I$ is $G$-stable, then so is ${\mathrm{lm}}(I)$.

Let $I$ be a $G$-stable ideal in $K[X]$. A [*$G$-Gröbner basis*]{} of $I$ is a subset $B$ of $I$ for which ${\mathrm{lm}}(GB)(=\{{\mathrm{lm}}(\pi
b) \mid b \in B,\ \pi  \in G\})$ generates the ideal ${\mathrm{lm}}(I)$ in ${\mathrm{Mon}}$. If $G$ is fixed in the context, then we also call $B$ an [*equivariant Gröbner basis*]{}. If $G=\{1\}$, then we call $B$ an [*ordinary Gröbner basis*]{}.

At MEGA 2009, 