The Simplify( G ) command attempts to simplify the presentation associated to the finitely presented group G. First, simplifications try to reduce the numbers of generators and relators, but this may cause an increase in the total relator length.

Use the tryhard option to force the Simplify command to employ more expensive simplifications that aim to reduce the total relator length.

You can use the PresentationComplexity command to measure the result of simplification.

The Simplify( G ) command employs several Tietze transformations, composed in a series of "rounds", to try to produce a "smaller" presentation for the input group G.  In the current implementation, each round consists of the following steps:

The first several simplifications listed are relatively inexpensive, and do not increase the total relator length, but may reduce the number of generators and relators.

Elimination by "long" relators is moderately expensive, and decreases the number of generators and relators, at the expense of an increase in the total relator length.

Common subword reduction is very expensive, and is only performed for presentations with few relators, or at the request of the user by the inclusion of the tryhard option.

Because the implementation of this command involves heuristics that employ a trade-off between the level of simplification that can be achieved as well as time- and space-performance, the implementation is expected to evolve.  As a result, it cannot be expected that the exact simplifications performed on any given finite presentation will be compatible between Maple releases.

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

$G≔⟨⟨a\,b\,c\,d⟩|⟨a^2\,b^2\,c^2\,d^2\,a·b=c\,b·c=d\,d·a=b⟩⟩$

$G≔⟨a\,b\,c\,d\mid a^2\,b^2\,c^2\,d^2\,a^{-1}{d}^{-1}b\,b^{-1}{a}^{-1}c\,c^{-1}{b}^{-1}d⟩$

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

$⟨a\mid a^2⟩$

$G≔⟨⟨a\,b\,c⟩|⟨a^2\,b^3\,c^5\,a·b·c=1⟩⟩$

$G≔⟨a\,b\,c\mid a^2\,abc\,b^3\,c^5⟩$

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

$3,4,13$

$G≔\mathrm{Simplify}\left(G\right)$

$G≔⟨a\,b\mid a^2\,b^3\,b^{-1}{a}^{-1}{b}^{-1}{a}^{-1}{b}^{-1}{a}^{-1}{b}^{-1}{a}^{-1}{b}^{-1}{a}^{-1}⟩$

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

$2,3,15$

$G≔\mathrm{SmallGroup}\left(32\,5\,'\mathrm{form}'=''fpgroup''\right)$

$G≔⟨\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\,\mathrm{a5}\mid {\mathrm{a2}}^2\,{\mathrm{a3}}^2\,{\mathrm{a5}}^2\,{\mathrm{a1}}^2{\mathrm{a4}}^{-1}\,{\mathrm{a4}}^2{\mathrm{a5}}^{-1}\,{\mathrm{a3}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a3}\mathrm{a1}\,{\mathrm{a3}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a3}\mathrm{a2}\,{\mathrm{a4}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a4}\mathrm{a1}\,{\mathrm{a4}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a4}\mathrm{a2}\,{\mathrm{a4}}^{-1}{\mathrm{a3}}^{-1}\mathrm{a4}\mathrm{a3}\,{\mathrm{a5}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a5}\mathrm{a1}\,{\mathrm{a5}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a5}\mathrm{a2}\,{\mathrm{a5}}^{-1}{\mathrm{a3}}^{-1}\mathrm{a5}\mathrm{a3}\,{\mathrm{a5}}^{-1}{\mathrm{a4}}^{-1}\mathrm{a5}\mathrm{a4}\,{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}{\mathrm{a3}}^{-1}⟩$

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

$5,15,53$

$G≔\mathrm{Simplify}\left(G\right)$

$G≔⟨\mathrm{a1}\,\mathrm{a2}\mid {\mathrm{a2}}^2\,{\mathrm{a1}}^{-2}{\mathrm{a2}}^{-1}{\mathrm{a1}}^2\mathrm{a2}\,{\mathrm{a1}}^8\,{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\,{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}\,{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}\,{\mathrm{a1}}^{-4}{\mathrm{a2}}^{-1}{\mathrm{a1}}^4\mathrm{a2}\,{\mathrm{a1}}^{-2}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^2{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\,{\mathrm{a1}}^{-4}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^4{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}⟩$

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

$2,9,74$

$G≔\mathrm{Simplify}\left(G\,'\mathrm{tryhard}'\right)$

$G≔⟨\mathrm{a1}\,\mathrm{a2}\mid {\mathrm{a2}}^2\,{\mathrm{a1}}^{-1}\mathrm{a2}{\mathrm{a1}}^2{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\,{\mathrm{a1}}^8\,{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}\mathrm{a2}\,{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}⟩$

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

$2,5,32$

$G≔⟨⟨a\,b⟩|⟨a^{10}·b·a·b=1\,a^{11}·b·a^2·b·a^{-11}·b=1⟩⟩\:$

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

$2,2,40$

$H≔\mathrm{Simplify}\left(G\right)$

$H≔⟨a\,b\mid a{b}^{-1}a{b}^{2}ab{a}^{-1}b\,a^{10}bab⟩$

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

$2,2,22$

$G≔⟨⟨a\,b\,c⟩|⟨a^3\,b^7\,c^{13}\,{\left(a·b\right)}^2\,{\left(b·c\right)}^2\,{\left(c·a\right)}^2\,{\left(a·b·c\right)}^2=1⟩⟩\:$

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

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

$2,60,25340$

$H≔\mathrm{Simplify}\left(H\right)\:$

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

$2,28,10795$

$H≔\mathrm{Simplify}\left(H\,'\mathrm{tryhard}'\right)\:$

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

$2,5,349$

The GroupTheory[Simplify] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.