The $p$-core of a group $G$ is the largest normal $p$-subgroup of $G$, where $p$ is a prime integer.  It is equal to the intersection of the Sylow $p$-subgroups of $G$, which is, in turn, equal to the core of any one Sylow $p$-subgroup.

The PCore( p, G ) command constructs the p-core of a group G. The group G must be an instance of a permutation group.

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

$G≔\mathrm{DihedralGroup}\left(14\right)$

$G≔{\mathbf{D}}_{14}$

$C≔\mathrm{PCore}\left(2\,G\right)$

$C≔⟨\left(1\,8\right)\left(2\,9\right)\left(3\,10\right)\left(4\,11\right)\left(5\,12\right)\left(6\,13\right)\left(7\,14\right)⟩$

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

$2$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(2\,G\right)\right)$

$4$

$C≔\mathrm{PCore}\left(7\,G\right)$

$C≔⟨\left(1\,3\,5\,7\,9\,11\,13\right)\left(2\,4\,6\,8\,10\,12\,14\right)⟩$

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

$7$

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

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