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.

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

$\mathrm{IsSimple}\left(\mathrm{CyclicGroup}\left(7\right)\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{Alt}\left(5\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{Alt}\left(n\right)\right)\mathbf{assuming}n::\mathrm{posint},4<n$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{PSL}\left(2\&comma;2\right)\right)$

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{PSL}\left(2\&comma;3\right)\right)$

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{PSL}\left(3\&comma;3\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{PGO}\left(-1\&comma;4\&comma;7\right)\right)$

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{PSO}\left(-1\&comma;4\&comma;7\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{PSL}\left(n\&comma;q\right)\right)\mathbf{assuming}q::\mathrm{primepower},3<q$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{HaradaNortonGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{DihedralGroup}\left(30\right)\right)$

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{Symm}\left(8\right)\right)$

$\mathrm{false}$

$\mathrm{IsSimple}\left(\mathrm{OrthogonalGroup}\left(''O8+(3)''\right)\right)$

$\mathrm{true}$

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

For more information on Maple 17 changes, see Updates in Maple 17.