If $H$ is a subgroup of a group $G$, then the core of $H$ in $G$ is the largest normal subgroup of $G$ contained in $H$.

The Core( H, G ) command computes the core of the subgroup H of the permutation group G.

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

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

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

$H≔\mathrm{Subgroup}\left(\left\{\left[\left[1\,5\,4\,3\,2\right]\right]\right\}\,G\right)$

$H≔⟨\left(1\,5\,4\,3\,2\right)⟩$

$C≔\mathrm{Core}\left(H\,G\right)$

$C≔⟨⟩$

$\mathrm{Generators}\left(C\right)$

$\mathrm{GroupOrder}\left(C\right)$

$1$

$G≔\mathrm{SymmetricGroup}\left(4\right)$

$G≔{\mathbf{S}}_4$

$H≔\mathrm{Subgroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,4\right]\,\left[2\,3\right]\right]\right)\right\}\,G\right)$

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

$C≔\mathrm{Core}\left(H\,G\right)$

$C≔⟨\left(1\,4\right)\left(2\,3\right)\,\left(1\,2\right)\left(3\,4\right)⟩$

$\mathrm{Generators}\left(C\right)$

$\left[\left(1\,4\right)\left(2\,3\right)\,\left(1\,2\right)\left(3\,4\right)\right]$

$\mathrm{GroupOrder}\left(C\right)$

$4$

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

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