Important: The group package has been deprecated. Use the superseding package GroupTheory instead.

This procedure attempts to find a set of relations among the given subgroup's generators sufficient to define the subgroup. The result is returned as a grelgroup.

The algorithm uses Todd-Coxeter coset enumeration, which is an inherently non-terminating process for infinite groups. Therefore, the algorithm will halt with an exception if too many cosets are generated during an attempt to enumerate cosets of a subgroup. The point at which the coset enumeration terminates is controlled by the environment variable _EnvMaxCosetsToddCoxeter, which has the default value $128000$.

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

$g≔\mathrm{grelgroup}\left(\left\{a\,b\,c\,d\right\}\,\left\{\left[a\,b\,c\,\frac{1}{d}\right]\,\left[b\,c\,d\,\frac{1}{a}\right]\,\left[c\,d\,a\,\frac{1}{b}\right]\,\left[d\,a\,b\,\frac{1}{c}\right]\right\}\right)\:$

$\mathrm{sg}≔\mathrm{subgrel}\left(\left\{x=\left[a\,b\right]\,y=\left[a\,c\right]\right\}\,g\right)\:$

$\mathrm{pres}\left(\mathrm{sg}\right)$

$\mathrm{grelgroup}\left(\left\{x\,y\right\}\,\left\{\left[\frac{1}{x}\,y\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,x\,\frac{1}{y}\right]\,\left[\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\right]\,\left[y\,\frac{1}{x}\,\frac{1}{x}\,\frac{1}{x}\,y\,x\,\frac{1}{y}\,x\,\frac{1}{y}\,x\right]\right\}\right)$

$s≔\mathrm{subgrel}\left(\left\{x=\left[a\,b\,\frac{1}{a}\right]\right\}\,\mathrm{grelgroup}\left(\left\{a\,b\right\}\,\left\{\left[b\,b\right]\,\left[a\,a\,a\right]\right\}\right)\right)\:$

$\mathrm{pres}\left(s\right)$

Error, (in group:-pres) too many cosets - subgroup may have infinite index; you can increase the coset limit by setting the environment variable _EnvMaxCosetsToddCoxeter to a value larger than 128000