Index - Maple Help

GroupTheory

 Index
 compute the index of a subgroup

 Calling Sequence Index( H, G )

Parameters

 G - a group H - a subgroup of G

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{Index}\left(\mathrm{Alt}\left(4\right),\mathrm{Alt}\left(5\right)\right)$
 ${5}$ (1)
 > $G≔⟨⟨a,b⟩|{a}^{3}={b}^{11}⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{-3}}{}{{b}}^{{11}}{}⟩$ (2)
 > $H≔\mathrm{Subgroup}\left(\left\{a·b,{a}^{2}\right\},G\right)$
 ${H}{≔}⟨{}{\mathrm{_G}}{,}{\mathrm{_G0}}{}{\mid }{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{,}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}{{\mathrm{_G}}}^{{-2}}{}{\mathrm{_G0}}{}{\mathrm{_G}}{}{\mathrm{_G0}}{}⟩$ (3)
 > $\mathrm{Index}\left(H,G\right)$
 ${2}$ (4)
 > $G≔⟨⟨x,y⟩|⟨{\left({y}^{2}\right)}^{x}={y}^{-2},{\left({x}^{2}\right)}^{y}={x}^{-2}⟩⟩:$
 > $S≔\mathrm{Subgroup}\left(\left\{{x}^{2},{y}^{2},{\left(x·y\right)}^{2}\right\},G\right):$
 > $\mathrm{Index}\left(S,G\right)$
 ${4}$ (5)

Compatibility

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