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.

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

$\mathrm{IsRegularPGroup}\left(\mathrm{SL}\left(4\,3\right)\right)$

$\mathrm{false}$

$\mathrm{IsRegularPGroup}\left(\mathrm{SmallGroup}\left(4\,1\right)\right)$

$\mathrm{true}$

$\mathrm{IsRegularPGroup}\left(\mathrm{SmallGroup}\left(4\,2\right)\right)$

$\mathrm{true}$

$\mathrm{IsRegularPGroup}\left(\mathrm{DihedralGroup}\left(32\right)\right)$

$\mathrm{false}$

$\mathrm{SearchSmallGroups}\left('\mathrm{pgroupprime}'=2\,'\mathrm{abelian}'=\mathrm{false}\,'\mathrm{regularpgroup}'=\mathrm{true}\right)$

$\mathrm{SearchSmallGroups}\left('\mathrm{pgroupprime}'=2\,'\mathrm{abelian}'=\mathrm{true}\,'\mathrm{regularpgroup}'=\mathrm{false}\right)$

$\mathrm{IsRegularPGroup}\left(\mathrm{SylowSubgroup}\left(2\,\mathrm{Symm}\left(4\right)\right)\right)$

$\mathrm{false}$

$\mathrm{IsRegularPGroup}\left(\mathrm{SylowSubgroup}\left(7\,\mathrm{Symm}\left(49\right)\right)\right)$

$\mathrm{false}$

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

$\mathrm{andmap}\left(\mathrm{IsRegularPGroup}\,L\right)$

$\mathrm{true}$

$\mathrm{Reg},\mathrm{Irr}≔\mathrm{selectremove}\left(\mathrm{IsRegularPGroup}\,\mathrm{AllSmallGroups}\left(81\right)\right)\:$

$\mathrm{nops}\left(\mathrm{Reg}\right),\mathrm{nops}\left(\mathrm{Irr}\right)$

$11,4$

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

$\mathrm{andmap}\left(\mathrm{IsRegularPGroup}\,L\right)$

$\mathrm{true}$

$G≔\mathrm{DirectProduct}\left(\mathrm{SearchSmallGroups}\left('\mathrm{order}'=125\,'\mathrm{regularpgroup}'\,'\mathrm{form}'=''permgroup''\right)\right)$

$G≔\mathrm{\⟨a\; permutation\; group\; on\; 625\; letters\; with\; 12\; generators\⟩}$

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

$\mathrm{true}$

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

For more information on Maple 2021 changes, see Updates in Maple 2021.