IsCaminaGroup - Maple Help

GroupTheory

 IsCaminaGroup
 determine whether a group is a Camina group

 Calling Sequence IsCaminaGroup( G )

Parameters

 G - a permutation group

Description

 • A non-abelian group $G$ is a Camina group if it is not perfect and if, for each $g\in G\setminus H$ we have ${g}^{G}=g·H$, where $H$ is the derived subgroup of $G$. That is, the conjugacy class of each element of $G$ not in the derived subgroup is equal to its coset of the derived subgroup. (In any group $G$, the conjugacy class ${g}^{G}$ of any element $g$ is contained in the coset $\mathrm{gH}$ of the derived subgroup.)
 • Examples of Camina groups are some (but not all) Frobenius groups and extraspecial $p$-groups, for prime numbers $p$.
 • The IsCaminaGroup( G ) command determines, for a permutation group G, whether G is a Camina group. It returns true if G is a Camina group, and returns false otherwise.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsCaminaGroup}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsCaminaGroup}\left(\mathrm{QuaternionGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsCaminaGroup}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsCaminaGroup}\left(\mathrm{DihedralGroup}\left(32\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsCaminaGroup}\left(\mathrm{SmallGroup}\left(72,41\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsCaminaGroup}\left(\mathrm{FrobeniusGroup}\left(968,2\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsCaminaGroup}\left(\mathrm{FrobeniusGroup}\left(300,1\right)\right)$
 ${\mathrm{false}}$ (9)
 > $G≔\mathrm{SmallGroup}\left(300,23\right):$
 > $\mathrm{IsCaminaGroup}\left(G\right)$
 ${\mathrm{false}}$ (10)
 > $\mathrm{IsFrobeniusGroup}\left(G\right)$
 ${\mathrm{true}}$ (11)
 > $H≔\mathrm{DerivedSubgroup}\left(G\right):$
 > $\mathrm{cc}≔\mathrm{remove}\left(g↦g\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}H,\mathrm{map}\left(\mathrm{Representative},\mathrm{ConjugacyClasses}\left(G\right)\right)\right):$
 > $\mathrm{nops}\left(\mathrm{cc}\right)$
 ${4}$ (12)
 > $\mathrm{nops}\left(\mathrm{remove}\left(g↦\mathrm{Elements}\left(\mathrm{ConjugacyClass}\left(g,G\right)\right)=\mathrm{Elements}\left(\mathrm{LeftCoset}\left(g,H\right)\right),\mathrm{cc}\right)\right)$
 ${2}$ (13)

Compatibility

 • The GroupTheory[IsCaminaGroup] command was introduced in Maple 2022.