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

$G:=\mathrm{PerfectGroup}\left(60\,1\right)$

$G:=⟨\left(2\,3\,4\right)\,\left(1\,2\right)\left(4\,5\right)⟩$

$\mathrm{IsPerfect}\left(G\right)$

$\mathrm{true}$

$\mathrm{NumPerfectGroups}\left(1920\right)$

$7$

$\mathrm{andmap}\left(\mathrm{IsPerfect}\,\mathrm{AllPerfectGroups}\left(1920\right)\right)$

$\mathrm{true}$

Many computations with finitely presented groups make use of a "coset enumeration" process, based on the Todd-Coxeter method. A new coset enumerator is being developed for the GroupTheory package, that is faster and more robust than the one in Maple 17.  In Maple 18, some commands (namely, Index and PermutationGroup,on a finitely presented group) make use of the new coset enumerator leading to better results in these cases.

$G:=n\→⟨ ⟨a\,b⟩ \| ⟨a^n\.b^{n\+1}\,a^{n\+1}\.b^{n\+2}\=1⟩ ⟩\:$

$\mathrm{PermutationGroup}\left(G\left(10000\right)\right)$

The new Simplify command implements a Tietze transformation program that can help simplify many group presentations by generators and defining relators.

An auxiliary command, PresentationComplexity, was also introduced to enable you to measure how complicated a finite presentation is.

$G:=\mathrm{SmallGroup}\left(192\,22\,\'\mathrm{form}\'\=''fpgroup''\right)$

$G:=⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a3}\,\mathrm{\_a4}\,\mathrm{\_a5}\,\mathrm{\_a6}\,\mathrm{\_a7}\mid {\mathrm{\_a6}}^2\,{\mathrm{\_a1}}^2{\mathrm{\_a3}}^{-1}\,{\mathrm{\_a2}}^2{\mathrm{\_a4}}^{-1}\,{\mathrm{\_a3}}^2{\mathrm{\_a5}}^{-1}\,{\mathrm{\_a4}}^2{\mathrm{\_a6}}^{-1}\,{\mathrm{\_a5}}^2{\mathrm{\_a6}}^{-1}\,{\mathrm{\_a7}}^3\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a4}\mathrm{\_a2}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a3}}^{-1}\mathrm{\_a4}\mathrm{\_a3}\,{\mathrm{\_a5}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a5}\mathrm{\_a1}\,{\mathrm{\_a5}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a5}\mathrm{\_a2}\,{\mathrm{\_a5}}^{-1}{\mathrm{\_a3}}^{-1}\mathrm{\_a5}\mathrm{\_a3}\,{\mathrm{\_a5}}^{-1}{\mathrm{\_a4}}^{-1}\mathrm{\_a5}\mathrm{\_a4}\,{\mathrm{\_a6}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a6}\mathrm{\_a1}\,{\mathrm{\_a6}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a6}\mathrm{\_a2}\,{\mathrm{\_a6}}^{-1}{\mathrm{\_a3}}^{-1}\mathrm{\_a6}\mathrm{\_a3}\,{\mathrm{\_a6}}^{-1}{\mathrm{\_a4}}^{-1}\mathrm{\_a6}\mathrm{\_a4}\,{\mathrm{\_a6}}^{-1}{\mathrm{\_a5}}^{-1}\mathrm{\_a6}\mathrm{\_a5}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a7}\mathrm{\_a2}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a3}}^{-1}\mathrm{\_a7}\mathrm{\_a3}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a4}}^{-1}\mathrm{\_a7}\mathrm{\_a4}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a5}}^{-1}\mathrm{\_a7}\mathrm{\_a5}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a6}}^{-1}\mathrm{\_a7}\mathrm{\_a6}\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}{\mathrm{\_a4}}^{-1}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a4}\mathrm{\_a1}{\mathrm{\_a6}}^{-1}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a7}\mathrm{\_a1}{\mathrm{\_a7}}^{-1}⟩$

$\mathrm{PresentationComplexity}\left(G\right)$

$7\,28\,107$

$H:=\mathrm{Simplify}\left(G\right)$

$H:=⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a7}\mid {\mathrm{\_a7}}^3\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a7}\mathrm{\_a2}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a7}\mathrm{\_a1}{\mathrm{\_a7}}^{-1}\,{\mathrm{\_a1}}^{-2}{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^2\mathrm{\_a2}\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}{\mathrm{\_a2}}^{-2}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a1}}^{-2}\mathrm{\_a7}{\mathrm{\_a1}}^2\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a2}}^{-2}\mathrm{\_a7}{\mathrm{\_a2}}^2\,{\mathrm{\_a2}}^8\,{\mathrm{\_a2}}^{-2}{\mathrm{\_a1}}^{-2}{\mathrm{\_a2}}^2{\mathrm{\_a1}}^2\,{\mathrm{\_a1}}^{-4}{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^4\mathrm{\_a2}\,{\mathrm{\_a2}}^{-2}{\mathrm{\_a1}}^{-1}{\mathrm{\_a2}}^2\mathrm{\_a1}{\mathrm{\_a2}}^{-4}\,{\mathrm{\_a2}}^{-4}{\mathrm{\_a1}}^{-1}{\mathrm{\_a2}}^4\mathrm{\_a1}\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a1}}^{-4}\mathrm{\_a7}{\mathrm{\_a1}}^4\,{\mathrm{\_a7}}^{-1}{\mathrm{\_a2}}^{-4}\mathrm{\_a7}{\mathrm{\_a2}}^4\,{\mathrm{\_a1}}^8{\mathrm{\_a2}}^{-4}\,{\mathrm{\_a1}}^{-4}{\mathrm{\_a2}}^{-2}{\mathrm{\_a1}}^4{\mathrm{\_a2}}^2\,{\mathrm{\_a2}}^{-4}{\mathrm{\_a1}}^{-2}{\mathrm{\_a2}}^4{\mathrm{\_a1}}^2\,{\mathrm{\_a2}}^{-4}{\mathrm{\_a1}}^{-4}{\mathrm{\_a2}}^4{\mathrm{\_a1}}^4⟩$

$\mathrm{PresentationComplexity}\left(H\right)$

$3\,18\,154$

The cycle index polynomial of a permutation group encodes the cycle structure of the permutations in the group as a multivariate polynomial that figures prominently in the Polya-Redfield theory of enumeration. The new CycleIndexPolynomial command computes the cycle index polynomial of a finite permutation group.

$\mathrm{CycleIndexPolynomial}\left(\mathrm{Symm}\left(3\right)\,\left[x\,y\,z\right]\right)$

$\frac{1}{6}{x}^3\+\frac{1}{2}y\+\frac{1}{3}z$

$G:=⟨ ⟨a\,b\,c⟩ \| ⟨a^2\=b^4\,a^6\,c^{-1}\.b\.c\=b^3⟩ ⟩$

$G:=⟨a\,b\,c\mid a^6\,a^{-2}{b}^4\,c^{-1}{b}^{-1}c{b}^3⟩$

$\mathrm{AbelianInvariants}\left(G\right)$

$\left[1\,\left[2\,2\right]\right]$