IsRegularPGroup - Maple Help

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GroupTheory

 IsRegularPGroup
 determine whether a group is regular a p-group, for some prime p

 Calling Sequence IsRegularPGroup( G )

Parameters

 G - : PermutationGroup : a $p$-group for some prime $p$.

Description

 • 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 ${a}^{{p}^{k}}·{b}^{{p}^{k}}={\left(a·b\right)}^{{p}^{k}}·{s}^{{p}^{k}}$, for some element $s$ in $\mathrm{Agemo}\left(1,H\right)$, 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsRegularPGroup}\left(\mathrm{SL}\left(4,3\right)\right)$
 ${\mathrm{false}}$ (1)

For $2$-groups, regularity is equivalent to commutativity.

 > $\mathrm{IsRegularPGroup}\left(\mathrm{SmallGroup}\left(4,1\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsRegularPGroup}\left(\mathrm{SmallGroup}\left(4,2\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsRegularPGroup}\left(\mathrm{DihedralGroup}\left(32\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{SearchSmallGroups}\left('\mathrm{pgroupprime}'=2,'\mathrm{abelian}'=\mathrm{false},'\mathrm{regularpgroup}'=\mathrm{true}\right)$
 > $\mathrm{SearchSmallGroups}\left('\mathrm{pgroupprime}'=2,'\mathrm{abelian}'=\mathrm{true},'\mathrm{regularpgroup}'=\mathrm{false}\right)$

The Sylow $2$-subgroup of ${\mathbf{S}}_{4}$ is a dihedral group of order $8$, so is non-abelian.

 > $\mathrm{IsRegularPGroup}\left(\mathrm{SylowSubgroup}\left(2,\mathrm{Symm}\left(4\right)\right)\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsRegularPGroup}\left(\mathrm{SylowSubgroup}\left(7,\mathrm{Symm}\left(49\right)\right)\right)$
 ${\mathrm{false}}$ (6)

Every group of order ${p}^{3}$, for odd primes $p$, is regular because they all have nilpotency class at most two.

 > $L≔\left[\mathrm{seq}\right]\left(\mathrm{SmallGroup}\left({7}^{3},k\right),k=1..\mathrm{NumGroups}\left({7}^{3}\right)\right):$
 > $\mathrm{andmap}\left(\mathrm{IsRegularPGroup},L\right)$
 ${\mathrm{true}}$ (7)

For $p=3$, there are irregular groups of order $81$.

 > $\mathrm{Reg},\mathrm{Irr}≔\mathrm{selectremove}\left(\mathrm{IsRegularPGroup},\mathrm{AllSmallGroups}\left(81\right)\right):$
 > $\mathrm{nops}\left(\mathrm{Reg}\right),\mathrm{nops}\left(\mathrm{Irr}\right)$
 ${11}{,}{4}$ (8)

However, for $3, the groups of order ${p}^{4}$ are all regular.

 > $L≔\left[\mathrm{seq}\right]\left(\mathrm{SmallGroup}\left({7}^{4},k\right),k=1..\mathrm{NumGroups}\left({7}^{4}\right)\right):$
 > $\mathrm{andmap}\left(\mathrm{IsRegularPGroup},L\right)$
 ${\mathrm{true}}$ (9)

Direct products of regular $p$-groups are regular.

 > $G≔\mathrm{DirectProduct}\left(\mathrm{SearchSmallGroups}\left('\mathrm{order}'=125,'\mathrm{regularpgroup}','\mathrm{form}'="permgroup"\right)\right)$
 ${G}{≔}{\mathrm{< a permutation group on 625 letters with 12 generators >}}$ (10)
 > $\mathrm{IsRegularPGroup}\left(G\right)$
 ${\mathrm{true}}$ (11)

Compatibility

 • The GroupTheory[IsRegularPGroup] command was introduced in Maple 2021.