determine whether a group is a p-group, for some prime p
determine the prime for which a group is a p-group
IsPGroup( G )
IsPGroup( G, prime = p )
PGroupPrime( G )
a prime number
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.
G ≔ Alt⁡4
Error, (in GroupTheory:-PGroupPrime) group does not have prime-power order
G ≔ DihedralGroup⁡8
The GroupTheory[IsPGroup] and GroupTheory[PGroupPrime] commands were introduced in Maple 2018.
For more information on Maple 2018 changes, see Updates in Maple 2018.
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