determine whether a group is regular a p-group, for some prime p
IsRegularPGroup( G )
: PermutationGroup : a p-group for some prime p.
A finite p-group G, where p is a prime, is said to be regular if, for any elements a and b in G, and for any positive integer k, we have apk·bpk=a·bpk·spk, for some element s in Agemo⁡1,H, where H is the derived subgroup of the subgroup of G generated by a and b.
For 2-groups, regularity is equivalent to commutativity.
Regularity as a p-group should not be confused with regularity as a permutation group. To test for regularity as a permutation group, see GroupTheory[IsRegular].
The IsRegularPGroup( G ) command returns true if the permutation group G is a regular p-group, for a prime number p, and returns false if it is not.
The Sylow 2-subgroup of S4 is a dihedral group of order 8, so is non-abelian.
Every group of order p3, for odd primes p, is regular because they all have nilpotency class at most two.
For p=3, there are irregular groups of order 81.
However, for 3<p, the groups of order p4 are all regular.
The GroupTheory[IsRegularPGroup] command was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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