Factor - Maple Help

GroupTheory

 Factor
 factor a group element into a subgroup element and a coset representative

 Calling Sequence Factor( g, H )

Parameters

 g - permutation or word on the generators of the supergroup of H H - a permutation group or a subgroup of a finitely presented group

Description

 • 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.

Examples

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

First we consider the following subgroup of the symmetric group of degree 7.

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

We can factor this permutation over the cosets of H in Symm(7).

 > $g≔\mathrm{Perm}\left(\left[\left[3,4,5,6\right]\right]\right)$
 ${g}{≔}\left({3}{,}{4}{,}{5}{,}{6}\right)$ (2)
 > $f≔\mathrm{Factor}\left(g,H\right)$
 ${f}{≔}\left[\left({3}{,}{4}{,}{5}{,}{7}{,}{6}\right){,}\left({6}{,}{7}\right)\right]$ (3)
 > $R≔\mathrm{map}\left(\mathrm{Representative},\mathrm{RightCosets}\left(H,\mathrm{Symm}\left(7\right)\right)\right)$
 ${R}{≔}\left\{\left({6}{,}{7}\right){,}\left(\right)\right\}$ (4)
 > $\mathrm{member}\left(f\left[2\right],R\right)$
 ${\mathrm{true}}$ (5)
 > $f\left[1\right]·f\left[2\right]$
 $\left({3}{,}{4}{,}{5}{,}{6}\right)$ (6)

Next, consider the group of the (2,3)-torus knot, which is an infinite group.

 > $G≔⟨⟨a,b⟩|{a}^{2}={b}^{3}⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{-2}}{}{{b}}^{{3}}{}⟩$ (7)

The following subgroup of G has index in G equal to 3.

 > $H≔\mathrm{Subgroup}\left(\left\{a,b·a·{b}^{-1},{b}^{-1}·a·b\right\},G\right)$
 ${H}{≔}⟨{}{\mathrm{_G}}{,}{\mathrm{_G0}}{,}{\mathrm{_G1}}{}{\mid }{}{{\mathrm{_G}}}^{{2}}{}{{\mathrm{_G0}}}^{{-2}}{,}{\mathrm{_G1}}{}{{\mathrm{_G0}}}^{{-2}}{}{\mathrm{_G1}}{}{{\mathrm{_G0}}}^{{-2}}{}{{\mathrm{_G}}}^{{2}}{}⟩$ (8)
 > $\mathrm{Factor}\left(b·a·b,H\right)$
 $\left[{{a}}^{{2}}{}{b}{}{{a}}^{{-1}}{}{{b}}^{{-1}}{}{{a}}^{{2}}{,}{{b}}^{{-1}}\right]$ (9)
 > $\mathrm{Factor}\left(a·b,H\right)$
 $\left[{{a}}^{{3}}{}{b}{}{{a}}^{{-2}}{}{{b}}^{{-1}}{,}{b}\right]$ (10)
 > $\mathrm{Factor}\left({\left(b·{a}^{2}\right)}^{2},H\right)$
 $\left[{{a}}^{{2}}{}{b}{}{{a}}^{{-1}}{}{{b}}^{{-1}}{}{{a}}^{{2}}{}{b}{}{{a}}^{{-1}}{}{{b}}^{{-1}}{}{{a}}^{{2}}{}{{b}}^{{-1}}{}{{a}}^{{2}}{}{b}{,}{{b}}^{{-1}}\right]$ (11)

The alternating group of degree 5 has the following presentation.

 > $G≔⟨⟨a,b⟩|⟨{a}^{2},{b}^{3},{\left(a·b\right)}^{5}=1⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{3}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}⟩$ (12)
 > $H≔\mathrm{Subgroup}\left(\left\{a·b\right\},G\right)$
 ${H}{≔}⟨{}{\mathrm{_G2}}{}{\mid }{}{{\mathrm{_G2}}}^{{5}}{}⟩$ (13)
 > $\mathrm{Factor}\left(a·b·b,H\right)$
 $\left[{a}{}{b}{,}{b}\right]$ (14)

Compatibility

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