GroupTheory/IsCharacteristicallySimple - Maple Help
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GroupTheory

 IsCharacteristicallySimple
 determine whether a group is characteristically simple

 Calling Sequence IsCharacteristicallySimple( G )

Parameters

 G - a group

Description

 • A group $G$ is characteristically simple if it is non-trivial and has no proper, non-trivial characteristic subgroups. Clearly, every simple group is characteristically simple.
 • The IsCharacteristicallySimple( G ) command returns true if the group G is characteristically simple, and returns false otherwise.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{CyclicGroup}\left(37\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{CyclicGroup}\left(300\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{Alt}\left(8\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{Symm}\left(8\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{SmallGroup}\left(8,5\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{SmallGroup}\left(8,2\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{DirectProduct}\left(\mathrm{Alt}\left(5\right),\mathrm{Alt}\left(6\right)\right)\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsCharacteristicallySimple}\left(\mathrm{DirectProduct}\left(\mathrm{Alt}\left(6\right),\mathrm{Alt}\left(6\right)\right)\right)$
 ${\mathrm{true}}$ (8)