The index of a subgroup $H$ of a group $G$ is the number of (left or right) cosets of $H$ in $G$.  If $G$ is finite, then the index of $H$ in $G$ is equal to $\frac{\left|G\right|}{\left|H\right|}$.

The Index( H, G ) command computes the index of the subgroup H of the group G.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{Index}\left(\mathrm{Alt}\left(4\right)\,\mathrm{Alt}\left(5\right)\right)$

$5$

$G≔⟨⟨a\,b⟩|a^3=b^{11}⟩$

$G≔⟨a\,b\mid a^{-3}{b}^{11}⟩$

$H≔\mathrm{Subgroup}\left(\left\{a·b\,a^2\right\}\,G\right)$

$H≔⟨\mathrm{\_G}\,\mathrm{\_G0}\mid {\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\,{\mathrm{\_G}}^{-2}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}{\mathrm{\_G}}^{-2}\mathrm{\_G0}\mathrm{\_G}\mathrm{\_G0}⟩$

$\mathrm{Index}\left(H\,G\right)$

$2$

$G≔⟨⟨x\,y⟩|⟨{\left(y^2\right)}^x=y^{-2}\,{\left(x^2\right)}^y=x^{-2}⟩⟩\:$

$S≔\mathrm{Subgroup}\left(\left\{x^2\,y^2\,{\left(x·y\right)}^2\right\}\,G\right)\:$

$\mathrm{Index}\left(S\,G\right)$

$4$

The GroupTheory[Index] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.