A group $G$ is a $p$-group, for a prime number $p$, if every member of $G$ has finite order equal to a power of $p$.

A finite group is a $p$-group if, and only if, its order is a power of $p$. A finite $p$-group is nilpotent.

The IsPGroup( G ) command attempts to determine whether the group G is a $p$-group, for some prime number $p$. It returns true if G is a $p$-group and returns false otherwise.

If the prime = p option is passed, with p an explicit prime number, then IsPGroup( G, prime = p ) checks whether G is a p-group. For example, to check whether G is a $3$-group, use the command IsPGroup( G, prime = 3 ).

The PGroupPrime( G ) command returns a prime number p if the group G is a non-trivial $p$-group.  If Maple can determine that G is a trivial group, then the value FAIL is returned (since the trivial group is a $p$-group, for all primes $p$, so the value is not well-defined).  If Maple can determine that G is not a $p$-group for any prime number $p$, then an exception is raised.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$\mathrm{IsPGroup}\left(G\right)$

$\mathrm{false}$

$\mathrm{PGroupPrime}\left(G\right)$

Error, (in GroupTheory:-PGroupPrime) group does not have prime-power order

$G≔\mathrm{DihedralGroup}\left(8\right)$

$G≔{\mathbf{D}}_8$

$\mathrm{IsPGroup}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsPGroup}\left(G\,'\mathrm{prime}'=3\right)$

$\mathrm{false}$

$\mathrm{PGroupPrime}\left(G\right)$

$2$

$\mathrm{IsPGroup}\left(\mathrm{QuasicyclicGroup}\left(17\right)\right)$

$\mathrm{true}$

$\mathrm{PGroupPrime}\left(\mathrm{QuasicyclicGroup}\left(17\right)\right)$

$17$

$\mathrm{IsPGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$

$\mathrm{false}$

$\mathrm{IsPGroup}\left(\mathrm{TrivialGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{PGroupPrime}\left(\mathrm{TrivialGroup}\left(\right)\right)$

$\mathrm{FAIL}$

The GroupTheory[IsPGroup] and GroupTheory[PGroupPrime] commands were introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.