IsPGroup - Maple Help

GroupTheory

 IsPGroup
 determine whether a group is a p-group, for some prime p
 PGroupPrime
 determine the prime for which a group is a p-group

 Calling Sequence IsPGroup( G ) IsPGroup( G, prime = p ) PGroupPrime( G )

Parameters

 G - a group p - a prime number

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{IsPGroup}\left(G\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{PGroupPrime}\left(G\right)$
 > $G≔\mathrm{DihedralGroup}\left(8\right)$
 ${G}{≔}{{\mathbf{D}}}_{{8}}$ (3)
 > $\mathrm{IsPGroup}\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsPGroup}\left(G,'\mathrm{prime}'=3\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{PGroupPrime}\left(G\right)$
 ${2}$ (6)
 > $\mathrm{IsPGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsPGroup}\left(\mathrm{TrivialGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{PGroupPrime}\left(\mathrm{TrivialGroup}\left(\right)\right)$
 ${\mathrm{FAIL}}$ (9)

Compatibility

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