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GroupTheory

  

MetacyclicGroup

  

construct a finite metacyclic group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

MetacyclicGroup(m, n, k)

MetacyclicGroup(m, n, k, s)

Parameters

m

-

a positive integer

n

-

a positive integer

k

-

a positive integer

s

-

(optional) equation of the form form= "fpgroup" or form = "permgroup" (default)

Description

• 

A group metacyclic if it has a cyclic normal subgroup the quotient by which is also cyclic. Every such group G can be generated by two elements a and b, with the subgroup a normal in G. The group G is then determined by the action of b on a. Since a is normal in G, it follows that the conjugate ab belongs to a so there is a positive integer k for which ab=ak. Thus, a finite metacyclic group G is completely determined by the orders of a and b and the integer k.

• 

The MetacyclicGroup( m, n, k ) command constructs a metacyclic group with generators a and b as described above, such that ab=ak, and where an=1 and bm=1.

• 

Note that the generators a and b need not have orders n and m, respectively, but that their orders are necessarily divisors of n and m.

• 

By default, a permutation group is returned, but you can create a finitely presented group by passing the 'form' = "fpgroup" option.

Examples

withGroupTheory:

MetacyclicGroup6,8,5

1,2,6,14,9,34,7,15,25,19,105,8,16,26,20,1112,17,27,36,31,2113,18,28,37,32,2223,29,38,44,41,3324,30,39,45,42,3435,40,46,48,47,43,1,4,12,23,35,24,13,52,7,17,29,40,30,18,83,10,21,33,43,34,22,116,15,27,38,46,39,28,169,19,31,41,47,42,32,2014,25,36,44,48,45,37,26

(1)

MetacyclicGroup6,8,5,form=permgroup

1,2,6,14,9,34,7,15,25,19,105,8,16,26,20,1112,17,27,36,31,2113,18,28,37,32,2223,29,38,44,41,3324,30,39,45,42,3435,40,46,48,47,43,1,4,12,23,35,24,13,52,7,17,29,40,30,18,83,10,21,33,43,34,22,116,15,27,38,46,39,28,169,19,31,41,47,42,32,2014,25,36,44,48,45,37,26

(2)

MetacyclicGroup6,8,5,form=fpgroup

a,ba6,b8,b-1aba5

(3)

In the following example, the first parameter 6 is a proper multiple of the order of the corresponding generator.

a,bopGeneratorsMetacyclicGroup6,8,4

a,b1,2,34,8,65,9,710,12,1411,13,1516,20,1817,21,1922,23,24,1,4,10,16,22,17,11,52,6,12,18,23,19,13,73,8,14,20,24,21,15,9

(4)

PermOrdera

3

(5)

PermOrderb

8

(6)

Compatibility

• 

The GroupTheory[MetacyclicGroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory[CyclicGroup]

GroupTheory[DicyclicGroup]