Core - Maple Help

GroupTheory

 Core
 construct the core of a subgroup of a group

 Calling Sequence Core( H, G )

Parameters

 H - a subgroup of G G - a permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,2,3,4,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}\right){,}\left({1}{,}{2}{,}{3}\right)⟩$ (1)
 > $H≔\mathrm{Subgroup}\left(\left\{\left[\left[1,5,4,3,2\right]\right]\right\},G\right)$
 ${H}{≔}⟨\left({1}{,}{5}{,}{4}{,}{3}{,}{2}\right)⟩$ (2)
 > $C≔\mathrm{Core}\left(H,G\right)$
 ${C}{≔}⟨⟩$ (3)
 > $\mathrm{Generators}\left(C\right)$
 $\left[\right]$ (4)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${1}$ (5)
 > $G≔\mathrm{SymmetricGroup}\left(4\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (6)
 > $H≔\mathrm{Subgroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,4\right],\left[2,3\right]\right]\right)\right\},G\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩$ (7)
 > $C≔\mathrm{Core}\left(H,G\right)$
 ${C}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩$ (8)
 > $\mathrm{Generators}\left(C\right)$
 $\left[\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)\right]$ (9)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${4}$ (10)

Compatibility

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