 GroupTheory - Maple Programming Help

Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/IsCCGroup

GroupTheory

 IsCCGroup
 determine whether a group is a (CC)-group
 IsCAGroup
 determine whether a group is a (CA)-group
 IsCNGroup
 determine whether a group is a (CN)-group

 Calling Sequence IsCCGroup( G ) IsCAGroup( G ) IsCNGroup( G )

Parameters

 G - a group

Description

 • 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 $\mathrm{PSL}\left(2,{2}^{n}\right)$, for $2. 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.

Examples

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

The symmetric group of degree $3$ is non-nilpotent and a (CC)-group, so it is, simultaneously, the smallest non-cyclic (CC)-group, the smallest non-Abelian (CA)-group, and the smallest non-nilpotent (CN)-group. (CN)-group.

 > $\left[\mathrm{IsNilpotent},\mathrm{IsCCGroup},\mathrm{IsCAGroup},\mathrm{IsCNGroup}\right]\left(\mathrm{Symm}\left(3\right)\right)$
 $\left[{\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}\right]$ (1)

The smallest non-Abelian group that is a (CA)-group, but is not a (CC)-group is the alternating group of degree $4$.

 > $\mathrm{IsCCGroup}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsCAGroup}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (3)

The smallest group that is a (CN)-group, but not a (CA)-group is the symmetric group of degree $4$.

 > $\mathrm{IsCAGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsCNGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (5)

The groups PSL( 2, 2^n ) are important examples of (CA)-groups.

 > $\mathrm{IsCAGroup}\left(\mathrm{PSL}\left(2,8\right)\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsCAGroup}\left(\mathrm{PSL}\left(2,9\right)\right)$
 ${\mathrm{false}}$ (7)

A Frobenius group need not be a (CA)-group.

 > $\mathrm{IsCAGroup}\left(\mathrm{FrobeniusGroup}\left(72,2\right)\right)$
 ${\mathrm{false}}$ (8)

The Suzuki groups $\mathrm{Sz}\left({2}^{2n+1}\right)$ are an important class of non-Abelian simple (CN)-groups.

 > $G≔\mathrm{ExceptionalGroup}\left("Sz\left(8\right)"\right)$
 ${G}{≔}{Sz}\left({8}\right)$ (9)
 > $\mathrm{IsCNGroup}\left(G\right)$
 ${\mathrm{true}}$ (10)

Compatibility

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