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

The cosocle of a group $G$ is the intersection of the maximal normal subgroups of $G$. It is also equal to the set of "normal non-generators" of $G$, that is, the set of elements of $G$ that can be omitted from any set $X$ for which $G$ is the normal closure of $X$.

The Socle( G ) command constructs the socle of a group G.

The Cosocle( G ) command constructs the cosocle of the group G.

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

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

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

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

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

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

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{Cosocle}\left(\mathrm{Symm}\left(4\right)\right)\,\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

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

$S≔{\mathbf{A}}_6$

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

$\mathrm{true}$

$G≔\mathrm{DirectProduct}\left(\mathrm{Alt}\left(5\right)\,\mathrm{Alt}\left(5\right)\right)$

$G≔⟨\left(1\,2\,3\,4\,5\right)\,\left(3\,4\,5\right)\,\left(6\,7\,8\,9\,10\right)\,\left(8\,9\,10\right)⟩$

$\mathrm{IsSubgroup}\left(G\,\mathrm{Socle}\left(G\right)\right)$

$\mathrm{true}$

$\mathrm{IsTrivial}\left(\mathrm{Cosocle}\left(\mathrm{Alt}\left(6\right)\right)\right)$

$\mathrm{true}$

$\mathrm{Cosocle}\left(\mathrm{CyclicGroup}\left(30\right)\right)$

$\mathrm{Cosocle}\left(\mathrm{CyclicGroup}\left(12\right)\right)$

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

The GroupTheory[Socle] and GroupTheory[Cosocle] commands were introduced in Maple 2019.

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