 Socle - Maple Help

GroupTheory

 Socle
 construct the socle of a group
 Cosocle
 construct the cosocle of a group Calling Sequence Socle( G ) Cosocle( 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 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. 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)
 > $\mathrm{AreIsomorphic}\left(\mathrm{Cosocle}\left(\mathrm{Symm}\left(4\right)\right),\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (4)
 > $S≔\mathrm{Socle}\left(\mathrm{Alt}\left(6\right)\right)$
 ${S}{≔}{{\mathbf{A}}}_{{6}}$ (5)
 > $\mathrm{IsSubgroup}\left(\mathrm{Alt}\left(6\right),S\right)$
 ${\mathrm{true}}$ (6)
 > $G≔\mathrm{DirectProduct}\left(\mathrm{Alt}\left(5\right),\mathrm{Alt}\left(5\right)\right)$
 ${G}{≔}{\mathrm{< a permutation group on 10 letters with 4 generators >}}$ (7)
 > $\mathrm{IsSubgroup}\left(G,\mathrm{Socle}\left(G\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsTrivial}\left(\mathrm{Cosocle}\left(\mathrm{Alt}\left(6\right)\right)\right)$
 ${\mathrm{true}}$ (9)

The cosocle of a cyclic group is trivial if, and only if, the group has square-free order.

 > $\mathrm{Cosocle}\left(\mathrm{CyclicGroup}\left(30\right)\right)$
 $⟨⟩$ (10)
 > $\mathrm{Cosocle}\left(\mathrm{CyclicGroup}\left(12\right)\right)$
 $⟨\left({1}{,}{7}\right)\left({2}{,}{8}\right)\left({3}{,}{9}\right)\left({4}{,}{10}\right)\left({5}{,}{11}\right)\left({6}{,}{12}\right)⟩$ (11) Compatibility

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