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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

Parameters

Description

Examples

Compatibility

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 PSL2&comma;2n, 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.

Examples

withGroupTheory&colon;

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.

IsNilpotent&comma;IsCCGroup&comma;IsCAGroup&comma;IsCNGroupSymm3

false&comma;true&comma;true&comma;true

(1)

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

IsCCGroupAlt4

false

(2)

IsCAGroupAlt4

true

(3)

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

IsCAGroupSymm4

false

(4)

IsCNGroupSymm4

true

(5)

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

IsCAGroupPSL2&comma;8

true

(6)

IsCAGroupPSL2&comma;9

false

(7)

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

IsCAGroupFrobeniusGroup72&comma;2

false

(8)

The Suzuki groups Sz22n+1 are an important class of non-Abelian simple (CN)-groups.

GExceptionalGroupSz(8)

GSz8

(9)

IsCNGroupG

true

(10)

Compatibility

• 

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.

See Also

GroupTheory

GroupTheory[IsAbelian]

GroupTheory[IsCyclic]

GroupTheory[IsNilpotent]

https://en.wikipedia.org/wiki/CA-group

https://en.wikipedia.org/wiki/CN-group