 IsExtraspecial - Maple Help

GroupTheory

 IsSpecial
 determine whether a group is a special p-group, for some prime p
 IsExtraspecial
 determine whether a group is an extraspecial p-group, for some prime p Calling Sequence IsSpecial( G ) IsExtraspecial( G ) Parameters

 G - : PermutationGroup : a permutation group Description

 • Let $G$ be a finite of prime-power order. We say that $G$ is special if either $G$ is elementary abelian, or if the center, derived subgroup, and Frattini subgroup of $G$ all coincide and is elementary abelian. If, in addition, these coindicent subgroups of $G$ have prime order, then $G$ is said to be extraspecial. Note that non-trivial abelian groups are not extraspecial, since their centers and derived subgroups cannot be equal.
 • The IsSpecial( G ) command returns true if the permutation group G is a special $p$-group, for some prime number $p$.
 • The IsExtraspecial( G ) command returns true if the permutation group G is an extraspecial $p$-group, for some prime number $p$.
 • Both commands return false if the group G is not a $p$-group for any prime number $p$. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsSpecial}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{IsSpecial}\left(\mathrm{CyclicGroup}\left(3\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsExtraspecial}\left(\mathrm{CyclicGroup}\left(3\right)\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{IsSpecial}\left(\mathrm{CyclicGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsSpecial}\left(\mathrm{ElementaryGroup}\left(11,4\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsExtraspecial}\left(\mathrm{ElementaryGroup}\left(11,4\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsSpecial}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsSpecial}\left(\mathrm{DihedralGroup}\left(16\right)\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{GroupOrder}\left(\mathrm{Center}\left(\mathrm{DihedralGroup}\left(16\right)\right)\right)$
 ${2}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{DihedralGroup}\left(16\right)\right)\right)$
 ${8}$ (10)
 > $\mathrm{IsSpecial}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (11)
 > $\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (12)
 > $\mathrm{map}\left(\mathrm{GroupOrder},\left[\mathrm{Center},\mathrm{DerivedSubgroup},\mathrm{FrattiniSubgroup}\right]\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$
 $\left[{2}{,}{2}{,}{2}\right]$ (13)
 > $\mathrm{IsSpecial}\left(\mathrm{SmallGroup}\left(1331,5\right)\right)$
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsSpecial}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$
 ${\mathrm{false}}$ (15)
 > $\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$
 ${\mathrm{false}}$ (16)
 > $\mathrm{GroupOrder}\left(\mathrm{Center}\left(\mathrm{QuaternionGroup}\left(5\right)\right)\right)$
 ${2}$ (17)
 > $\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{QuaternionGroup}\left(5\right)\right)\right)$
 ${8}$ (18)
 > $\mathrm{IsExtraspecial}\left(\mathrm{SmallGroup}\left(1331,5\right)\right)$
 ${\mathrm{false}}$ (19)
 > $\mathrm{IsSpecial}\left(\mathrm{TrivialGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (20)
 > $\mathrm{IsExtraspecial}\left(\mathrm{TrivialGroup}\left(\right)\right)$
 ${\mathrm{false}}$ (21)
 > $G≔\mathrm{PSL}\left(2,9\right):$
 > $g≔\mathrm{RandomInvolution}\left(G\right):$
 > $C≔\mathrm{Centralizer}\left(g,G\right)$
 ${C}{≔}⟨\left({1}{,}{5}\right)\left({3}{,}{6}\right){,}\left({2}{,}{4}\right)\left({5}{,}{6}\right){,}\left({1}{,}{3}\right)\left({5}{,}{6}\right)⟩$ (22)
 > $\mathrm{IsExtraspecial}\left(C\right)$
 ${\mathrm{true}}$ (23)
 > $\mathrm{AreIsomorphic}\left(C,\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{true}}$ (24)