A group $G$ is a (CC)-group if the centralizer of each of its non-trivial elements is cyclic. Cyclic groups are obviously (CC)-groups, but there are non-Abelian (CC)-groups, such as the symmetric group of degree $3$.

A group $G$ is a (CA)-group if the centralizer of each of its non-trivial elements is Abelian. This is equivalent to commutativity being a transitive relation on the elements of the group. The only non-Abelian finite simple groups that are (CA)-groups are the groups $\mathrm{PSL}\left(2\&comma;2^n\right)$, for $2<n$. In fact, a (CA)-group is either soluble or simple, and a non-simple, non-Abelian finite (CA)-group is a Frobenius group.

A group $G$ is a (CN)-group if the centralizer of each of its non-trivial elements is nilpotent.

The classes of (CA)-groups and (CN)-groups were important in the development of the classification of the finite simple groups. The proof, by Suzuki (1957), that a finite simple non-Abelian (CA)-group has even order, and the proof, by Feit-Hall-Thompson (1960), that the same is true for (CN)-groups, presaged the later proof of the Odd-Order Theorem, by Feit and Thompson (1963).

The IsCCGroup( G ) command returns true if the group G is a (CC)-group and returns false otherwise.

The IsCAGroup( G ) command returns true if the group G is a (CA)-group and returns false otherwise.

The IsCNGroup( G ) command returns true if the group G is a (CN)-group and returns false otherwise.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$\left[\mathrm{IsNilpotent}\&comma;\mathrm{IsCCGroup}\&comma;\mathrm{IsCAGroup}\&comma;\mathrm{IsCNGroup}\right]\left(\mathrm{Symm}\left(3\right)\right)$

$\left[\mathrm{false}\&comma;\mathrm{true}\&comma;\mathrm{true}\&comma;\mathrm{true}\right]$

$\mathrm{IsCCGroup}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsCAGroup}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsCAGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsCNGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsCAGroup}\left(\mathrm{PSL}\left(2\&comma;8\right)\right)$

$\mathrm{true}$

$\mathrm{IsCAGroup}\left(\mathrm{PSL}\left(2\&comma;9\right)\right)$

$\mathrm{false}$

$\mathrm{IsCAGroup}\left(\mathrm{FrobeniusGroup}\left(72\&comma;2\right)\right)$

$\mathrm{false}$

$G≔\mathrm{ExceptionalGroup}\left(''Sz(8)''\right)$

$G≔Sz\left(8\right)$

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

$\mathrm{true}$

The GroupTheory[IsCCGroup], GroupTheory[IsCAGroup] and GroupTheory[IsCNGroup] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.