IsSimple - Maple Help

GroupTheory

 IsSimple
 determine whether a group is simple

 Calling Sequence IsSimple( G )

Parameters

 G - a group

Description

 • A group $G$ is simple if it has at least two members, and the only normal subgroups of $G$ are $G$ and the trivial subgroup. Alternatively, the group $G$ is simple if it has no proper homomorphic images.
 • The IsSimple( G ) command returns true if the group G is simple, and returns false otherwise.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsSimple}\left(\mathrm{CyclicGroup}\left(7\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsSimple}\left(\mathrm{CyclicGroup}\left(12\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsSimple}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{IsSimple}\left(\mathrm{Alt}\left(5\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsSimple}\left(\mathrm{Alt}\left(n\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}n::'\mathrm{posint}',4
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsSimple}\left(\mathrm{PSL}\left(2,2\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsSimple}\left(\mathrm{PSL}\left(2,3\right)\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsSimple}\left(\mathrm{PSL}\left(3,3\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsSimple}\left(\mathrm{PSL}\left(n,q\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}q::\mathrm{primepower},3
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsSimple}\left(\mathrm{HaradaNortonGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{IsSimple}\left(\mathrm{DihedralGroup}\left(30\right)\right)$
 ${\mathrm{false}}$ (11)
 > $\mathrm{IsSimple}\left(\mathrm{Symm}\left(8\right)\right)$
 ${\mathrm{false}}$ (12)
 > $\mathrm{IsSimple}\left(\mathrm{OrthogonalGroup}\left("O8+\left(3\right)"\right)\right)$
 ${\mathrm{true}}$ (13)

Compatibility

 • The GroupTheory[IsSimple] command was introduced in Maple 17.