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.

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

$\mathrm{IsCaminaGroup}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsCaminaGroup}\left(\mathrm{QuaternionGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsExtraspecial}\left(\mathrm{QuaternionGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsCaminaGroup}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{IsCaminaGroup}\left(\mathrm{SmallGroup}\left(72\,41\right)\right)$

$\mathrm{true}$

$\mathrm{IsCaminaGroup}\left(\mathrm{FrobeniusGroup}\left(968\,2\right)\right)$

$\mathrm{true}$

$\mathrm{IsCaminaGroup}\left(\mathrm{FrobeniusGroup}\left(300\,1\right)\right)$

$\mathrm{false}$

$G≔\mathrm{SmallGroup}\left(300\,23\right)\:$

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

$\mathrm{false}$

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

$\mathrm{true}$

$H≔\mathrm{DerivedSubgroup}\left(G\right)\:$

$\mathrm{cc}≔\mathrm{remove}\left(g\mapsto g\mathbf{in}H\,\mathrm{map}\left(\mathrm{Representative}\,\mathrm{ConjugacyClasses}\left(G\right)\right)\right)\:$

$\mathrm{nops}\left(\mathrm{cc}\right)$

$4$

$\mathrm{nops}\left(\mathrm{remove}\left(g\mapsto \mathrm{Elements}\left(\mathrm{ConjugacyClass}\left(g\,G\right)\right)=\mathrm{Elements}\left(\mathrm{LeftCoset}\left(g\,H\right)\right)\,\mathrm{cc}\right)\right)$

$2$

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

For more information on Maple 2022 changes, see Updates in Maple 2022.