Let $H$ be a subgroup of a group $G$, and let $g$ be a member of $G$. Let $R$ be a complete set of representatives of the right cosets of $H$ in $G$.  Then $g$ can be written, uniquely, in the form $g=h·r$, with $h$ in $H$ and $r$ in $R$.

The Factor( g, H ) command returns a pair [ h, r ], where h belongs to H, and r is a coset representative for the coset H.g in a supergroup of H.

If H is a permutation group, then the representative is for the cosets of H in the full symmetric group of the same degree as H. If H is a subgroup of a finitely presented group G, then the representative r is for the cosets of H in G.

The set of representatives used is the set obtained from the RightCosets command applied to H.

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

$H≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[3\,4\,5\,6\,7\right]\right]\right)\right)$

$H≔⟨\left(1\,2\,3\right)\,\left(3\,4\,5\,6\,7\right)⟩$

$g≔\mathrm{Perm}\left(\left[\left[3\,4\,5\,6\right]\right]\right)$

$g≔\left(3\,4\,5\,6\right)$

$f≔\mathrm{Factor}\left(g\,H\right)$

$f≔\left[\left(3\,4\,5\,7\,6\right)\,\left(6\,7\right)\right]$

$R≔\mathrm{map}\left(\mathrm{Representative}\,\mathrm{RightCosets}\left(H\,\mathrm{Symm}\left(7\right)\right)\right)$

$R≔\left\{\left(6\,7\right)\,\left(\right)\right\}$

$\mathrm{member}\left(f\left[2\right]\,R\right)$

$\mathrm{true}$

$f\left[1\right]·f\left[2\right]$

$\left(3\,4\,5\,6\right)$

$G≔⟨⟨a\,b⟩|a^2=b^3⟩$

$G≔⟨a\,b\mid a^{-2}{b}^3⟩$

$H≔\mathrm{Subgroup}\left(\left\{a\,b·a·b^{-1}\,b^{-1}·a·b\right\}\,G\right)$

$H≔⟨\mathrm{\_G}\,\mathrm{\_G0}\,\mathrm{\_G1}\mid {\mathrm{\_G}}^2{\mathrm{\_G0}}^{-2}\,\mathrm{\_G1}{\mathrm{\_G0}}^{-2}\mathrm{\_G1}{\mathrm{\_G0}}^{-2}{\mathrm{\_G}}^2⟩$

$\mathrm{Factor}\left(b·a·b\,H\right)$

$\left[a^{2}b{a}^{-1}{b}^{-1}{a}^2\,b^{-1}\right]$

$\mathrm{Factor}\left(a·b\,H\right)$

$\left[a^{3}b{a}^{-2}{b}^{-1}\,b\right]$

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

$\left[a^{2}b{a}^{-1}{b}^{-1}{a}^{2}b{a}^{-1}{b}^{-1}{a}^2{b}^{-1}{a}^{2}b\,b^{-1}\right]$

$G≔⟨⟨a\,b⟩|⟨a^2\,b^3\,{\left(a·b\right)}^5=1⟩⟩$

$G≔⟨a\,b\mid a^2\,b^3\,ababababab⟩$

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

$H≔⟨\mathrm{\_G2}\mid {\mathrm{\_G2}}^5⟩$

$\mathrm{Factor}\left(a·b·b\,H\right)$

$\left[ab\,b\right]$

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

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