Socle - Maple Help

GroupTheory

 Socle
 construct the socle of a group

 Calling Sequence Socle( G )

Parameters

 G - a permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $S≔\mathrm{Socle}\left(\mathrm{Symm}\left(4\right)\right)$
 ${S}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩$ (1)
 > $\mathrm{df}≔\left[\mathrm{DirectFactors}\right]\left(S\right)$
 ${\mathrm{df}}{≔}\left[⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩{,}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩\right]$ (2)
 > $\mathrm{andmap}\left(\mathrm{IsSimple},\mathrm{df}\right)$
 ${\mathrm{true}}$ (3)
 > $S≔\mathrm{Socle}\left(\mathrm{Alt}\left(6\right)\right)$
 ${S}{≔}{{\mathbf{A}}}_{{6}}$ (4)
 > $\mathrm{IsSubgroup}\left(\mathrm{Alt}\left(6\right),S\right)$
 ${\mathrm{true}}$ (5)

Compatibility

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