The socle of a group $G$ is the subgroup generated by the minimal normal (non-trivial) subgroups of $G$.

The Socle( G ) command constructs the socle of a group G. The group G must be an instance of a permutation group.

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

$S≔\mathrm{Socle}\left(\mathrm{Symm}\left(4\right)\right)$

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

$\mathrm{df}≔\left[\mathrm{DirectFactors}\right]\left(S\right)$

$\mathrm{df}≔\left[⟨\left(1\,2\right)\left(3\,4\right)⟩\,⟨\left(1\,4\right)\left(2\,3\right)⟩\right]$

$\mathrm{andmap}\left(\mathrm{IsSimple}\,\mathrm{df}\right)$

$\mathrm{true}$

$S≔\mathrm{Socle}\left(\mathrm{Alt}\left(6\right)\right)$

$S≔{\mathbf{A}}_6$

$\mathrm{IsSubgroup}\left(\mathrm{Alt}\left(6\right)\,S\right)$

$\mathrm{true}$

The GroupTheory[Socle] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.